Codes: An Introduction to Information Communication and Cryptography [1 ed.] 1848002726, 9781848002722

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Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain University of Dundee K. Erdmann University of Oxford A. MacIntyre Queen Mary, University of London L.C.G. Rogers University of Cambridge E. S¨uli University of Oxford J.F. Toland University of Bath

For other titles published in this series, go to www.springer.com/series/3423

Norman L. Biggs

Codes: An Introduction to Information Communication and Cryptography

123

Norman L. Biggs Department of Mathematics London School of Economics Houghton Street London WC2A 2AE, UK

Maple is a trademark of Waterloo Maple Inc.

Springer Undergraduate Mathematics Series ISSN 1615-2085

ISBN: 978-1-84800-272-2 DOI: 10.1007/978-1-84800-273-9

e-ISBN: 978-1-84800-273-9

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008930146 Mathematics Subject Classification (2000): 94A, 94B, 11T71 c Springer-Verlag London Limited 2008  Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 Springer Science+Business Media springer.com

Preface

Many people do not realise that mathematics provides the foundation for the devices we use to handle information in the modern world. Most of those who do know probably think that the parts of mathematics involved are quite ‘classical’, such as Fourier analysis and differential equations. In fact, a great deal of the mathematical background is part of what used to be called ‘pure’ mathematics, indicating that it was created in order to deal with problems that originated within mathematics itself. It has taken many years for mathematicians to come to terms with this situation, and some of them are still not entirely happy about it. This book is an integrated introduction to Coding. By this I mean replacing symbolic information, such as a sequence of bits or a message written in a natural language, by another message using (possibly) different symbols. There are three main reasons for doing this: Economy (data compression), Reliability (correction of errors), and Security (cryptography). I have tried to cover each of these three areas in sufficient depth so that the reader can grasp the basic problems and go on to more advanced study. The mathematical theory is introduced in a way that enables the basic problems to be stated carefully, but without unnecessary abstraction. The prerequisites (sets and functions, matrices, finite probability) should be familiar to anyone who has taken a standard course in mathematical methods or discrete mathematics. A course in elementary abstract algebra and/or number theory would be helpful, but the book contains the essential facts, and readers without this background should be able to understand what is going on.

vi

There are a few places where reference is made to computer algebra systems. I have tried to avoid making this a prerequisite, but students who have access to such a system will find it helpful. In particular, there are occasional specific references to MAPLETM (release 10), by Maplesoft, a division of Waterloo Maple Inc., Waterloo, Canada. The book has been developed from a course of twenty lectures on Information, Communication, and Cryptography given for the MSc in Applicable Mathematics at the London School of Economics. I should like to thank all those students who have contributed to the development of the course materials, in particular those who have written dissertations in this area: Rajni Kanda, Ovijit Paul, Arunduti Dutta-Roy, Ana de Corbavia-Perisic, Raminder Ruprai, James Rees, Elisabeth Biell, Anisa Bhatt, Timothy Morill, Shivam Kumar, and Carey Chua. I owe a special debt to Raminder Ruprai, who worked through all the exercises and helped to sort out many mistakes and obscurities. Finally, I am grateful to Aaron Wilson, who helped to produce the diagrams, and especially to Karen Borthwick, who has been very helpful and supportive on behalf of the publishers. Norman Biggs January 2008

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

1.

Coding and its uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Coding for economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Coding for reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Coding for security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 7 8 9

2.

Prefix-free codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The decoding problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Representing codes by trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Kraft-McMillan number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Unique decodability implies K ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Proof of the Counting Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 16 18 21 24

3.

Economical coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The concept of a source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Entropy, uncertainty, and information . . . . . . . . . . . . . . . . . . . . . . . 3.5 Optimal codes – the fundamental theorems . . . . . . . . . . . . . . . . . . 3.6 Huffman’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Optimality of Huffman codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 30 32 34 38 40 44

viii

4.

Data compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Coding in blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Distributions on product sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stationary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Coding a stationary source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Algorithms for data compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Using numbers as codewords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Arithmetic coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The properties of arithmetic coding . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Coding with a dynamic dictionary . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 49 52 55 58 59 62 65 67

5.

Noisy channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The definition of a channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Transmitting a source through a channel . . . . . . . . . . . . . . . . . . . . 5.3 Conditional entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The capacity of a channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Calculating the capacity of a channel . . . . . . . . . . . . . . . . . . . . . . .

73 73 76 78 81 83

6.

The problem of reliable communication . . . . . . . . . . . . . . . . . . . . . 89 6.1 Communication using a noisy channel . . . . . . . . . . . . . . . . . . . . . . . 89 6.2 The extended BSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3 Decision rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.4 Error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.5 The packing bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.

The noisy coding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.1 The probability of a mistake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 Coding at a given rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.3 Transmission using the extended BSC . . . . . . . . . . . . . . . . . . . . . . . 113 7.4 The rate should not exceed the capacity . . . . . . . . . . . . . . . . . . . . . 117 7.5 Shannon’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.6 Proof of Fano’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.

Linear codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.1 Introduction to linear codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Construction of linear codes using matrices . . . . . . . . . . . . . . . . . . 126 8.3 The check matrix of a linear code . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.4 Constructing 1-error-correcting codes . . . . . . . . . . . . . . . . . . . . . . . 131 8.5 The decoding problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Contents

9.

ix

Algebraic coding theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.1 Hamming codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.2 Cyclic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.3 Classification and properties of cyclic codes . . . . . . . . . . . . . . . . . . 149 9.4 Codes that can correct more than one error . . . . . . . . . . . . . . . . . . 153 9.5 Definition of a family of BCH codes . . . . . . . . . . . . . . . . . . . . . . . . . 155 9.6 Properties of the BCH codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

10. Coding natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.1 Natural languages as sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 The uncertainty of english . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 10.3 Redundancy and meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.4 Introduction to cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.5 Frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11. The development of cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.1 Symmetric key cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.2 Poly-alphabetic encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 11.3 The Playfair system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.4 Mathematical algorithms in cryptography . . . . . . . . . . . . . . . . . . . 185 11.5 Methods of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 12. Cryptography in theory and practice . . . . . . . . . . . . . . . . . . . . . . . 191 12.1 Encryption in terms of a channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 12.2 Perfect secrecy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 12.3 The one-time pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 12.4 Iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 12.5 Encryption standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 12.6 The key distribution problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 13. The RSA cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 13.1 A new approach to cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 13.2 Outline of the RSA system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 13.3 Feasibility of RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 13.4 Correctness of RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13.5 Confidentiality of RSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 14. Cryptography and calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 14.1 The scope of cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 14.2 Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 14.3 Calculations in the field Fp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 14.4 The discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

x

14.5 The ElGamal cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 14.6 The Diffie-Hellman key distribution system . . . . . . . . . . . . . . . . . . 230 14.7 Signature schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 15. Elliptic curve cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 15.1 Calculations in finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 15.2 The general ElGamal cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . . 239 15.3 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 15.4 The group of an elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 15.5 Improving the efficiency of exponentiation . . . . . . . . . . . . . . . . . . . 248 15.6 A final word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Answers to odd-numbered exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

1 Coding and its uses

1.1 Messages The first task is to set up a simple mathematical model of a message. We do this by looking at some examples and extracting some common features from them.

Example 1.1 Many messages are written in a natural language, such as English. These messages contain symbols, and the symbols form words, which in turn form sentences, such as this one. The messages may be sent from one person to another in several ways: in the form of a handwritten note or an email, for example. A text message is essentially the same, but it is often expressed in an unnatural language.

Example 1.2 Devices such as scanners and digital cameras produce messages in the form of electronic impulses. These messages may be sent from one device to another by wires or optic fibres, or by radio waves. Formal definitions based on these examples will be given in Section 1.3. For the time being, we shall think of a message as a sequence of symbols, noting N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 1, 

2

1. Coding and its uses

that the order of the symbols is clearly important. The function of a message is to convey information from a sender to a receiver. In order to do this successfully, the sender and receiver must agree to use the same set of symbols. This set is called an alphabet.

Example 1.3 We denote by A the alphabet which has 27 symbols, the letters A, B, C, . . ., Z, and a ‘space’, which we denote by ⊔. We shall often use the alphabet A to represent messages written in English. This is convenient for the sake of exposition, but obviously some features are ignored. Thus we ignore the distinction between upper and lower case letters, and we omit punctuation marks. Of course, there may be some loss in reducing an English message into a string of symbols in this alphabet. For example the text The word ‘hopefully’ is often misused. is reduced to the following message in A. THE⊔WORD⊔HOPEFULLY⊔IS⊔OFTEN⊔MISUSED

Example 1.4 The alphabet B has 2 symbols, 0 and 1, which are called binary digits or bits. Because the bits 0 and 1 can be implemented electronically as the states off and on, this is the underlying alphabet for all modern applications. In practice, the bits are often combined into larger groups, such as ‘32-bit words’. But any message that is transmitted electronically, whether it originates as an email from me or as an image from a satellite orbiting the earth, is essentially a sequence of bits.

EXERCISES 1.1. The following messages have been translated from ‘proper English’ into the alphabet A. Write down the original messages and comment upon any ambiguity or loss of meaning that has occurred. CANINE⊔HAS⊔SIX⊔LETTERS⊔AND⊔ENDS⊔IN⊔NINE ITS⊔HOT⊔SAID⊔ROBERT⊔BROWNING 1.2. A 32-bit word is a sequence of 32 symbols from the alphabet B. How many different 32-bit words are there? If my printer can print one every second, how many years (approximately) will it take to print them all?

1.2 Coding

3

1.3. In the period 1967-86 the ASCII alphabet was widely used as a standard for electronic communication. It has 128 symbols, 95 of which were printable. In this book we have already used some symbols that were not in the ASCII alphabet. Which ones? [ASCII is an abbreviation for American Standard Code for Information Interchange, and is pronounced ‘askey’. The ASCII alphabet is now part of a much more comprehensive system known as Unicode.] 1.4. Not all natural languages use 26 letters. How many letters are there in (i) the modern Greek alphabet and (ii) the Russian Cyrillic alphabet?

1.2 Coding Roughly speaking, coding is a rule for replacing one message by another message. The second message may or may not use the same alphabet as the first.

Example 1.5 A simple rule for coding messages in the 27-symbol alphabet A using the same alphabet is: write each word backwards. So the message SEE⊔YOU⊔TOMORROW

becomes

.

EES⊔UOY⊔WORROMOT

Example 1.6 A rule for coding messages in A using the binary alphabet B is: replace vowels by 0, replace consonants by 1, and ignore the spaces. With this rule SEE⊔YOU⊔TOMORROW

becomes

10010010101101

.

These two examples are very artificial, and the rules are of limited value. For greater realism and utility we must look at the purposes for which coding is used, and evaluate proposed coding rules in that context. There are three major reasons for coding a message. Economy In many situations it is necessary or desirable to use an alphabet smaller than those that occur in natural languages. It may also be desirable to make the message itself smaller: in recent times this has led to the development of techniques for Data Compression.

4

1. Coding and its uses

Reliability Messages may be altered by ‘noise’ in the process of transmission. Thus there is a need for codes that allow for Error Correction. Security Some messages are sent with the requirement that only the right person can understand them. Historically, secrecy was needed mainly in diplomatic and military communications, but nowadays it plays an important part in everyday commercial transactions. This area of coding is known as Cryptography.

EXERCISES 1.5. The following messages are coded versions of meaningful English sentences. Explain the coding rules used and find the original messages. 7 15 15 4 27 12 21 3 11 00111 01111 01111 00100 11011 01100 10101 00011 01011 1.6. Explain formally (as if you were writing a computer program) the coding rule write each word backwards. [You must explain how to convert a sequence of symbols such as TODAY⊔IS⊔MONDAY into YADOT⊔SI⊔YADNOM.]

1.3 Basic definitions We are now ready to make some proper definitions.

Definition 1.7 (Alphabet) An alphabet is a finite set S; we shall refer to the members of S as symbols.

Definition 1.8 (Message, string, word) A message in the alphabet S is a finite sequence of members of S: x1 x2 · · · xn

(xi ∈ S, 1 ≤ i ≤ n).

A message is often referred to as a string of members of S, or a word in S. The number n is called the length of the message, string, or word.

1.3 Basic definitions

5

The set of all strings of length n is denoted by S n . For example, when S = B and n = 3, the set B3 consists of the strings 000 001 010 011 100 101 110 111

.

The set of all strings in S is denoted by S ∗ : S∗ = S0 ∪ S1 ∪ S2 ∪ · · ·

.

Note that S 0 consists of the string with length zero; in other words, the string with no symbols. We include it in the definition because sometimes it is convenient to use it.

Definition 1.9 (Code, codeword) Let S and T be alphabets. A code c for S using T is an injective function c : S → T ∗ . For each symbol s ∈ S the string c(s) ∈ T ∗ is called the codeword for s. The set of all codewords, C = {c(s) | s ∈ S}, is also referred to as the code. When |T | = 2 the code is said to be binary, when |T | = 3 it is ternary, and in general when |T | = b, it is b-ary. For example, let S = {x, y, z}, T = B, and define c(x) = 0,

c(y) = 10,

c(z) = 11.

This is a binary code, and the set of codewords is C = {0, 10, 11}. According to the definition, a code c assigns to each symbol in S a string of symbols in T . The strings may vary in length. For example, suppose we are trying to construct a code for the 27-symbol English alphabet A using the binary alphabet B. We might begin by choosing codewords of length 4, as follows: A → 0000 B → 0001 C → 0010 . . . . Now, the definition requires c to be an injective function or (as we usually say) an injection. This is the mathematical form of the very reasonable requirement that c does not assign the same codeword to two different symbols. In other words, if c(s) = c(s′ ) then s = s′ . Clearly, there are only 16 strings of length 4 in B, so the 27 symbols in A cannot all be assigned different ones. Thus far we have considered only the coding of individual symbols. The extension to messages (strings of symbols) is clear.

6

1. Coding and its uses

Definition 1.10 (Concatenation) If c : S → T ∗ is a code, we extend c to S ∗ as follows. Given a string x1 x2 · · · xn in S ∗ , define c(x1 x2 · · · xn ) = c(x1 )c(x2 ) · · · c(xn ). This process is known as concatenation. Note that we denote the extended function S ∗ → T ∗ by the same letter c. It is not always possible to recover the original string uniquely from the coded version. For example, let S = {x, y, z}, and define c : S → B∗ by x → 0,

y → 01,

z → 10.

Suppose we are given the string 010100 which, we are told, is the result of coding a string in S ∗ using c. By trial and error we find two possibilities (at least): xzzx → 0 10 10 0 , yyxx → 01 01 0 0 . Clearly, this situation is to be avoided, if possible.

Definition 1.11 (Uniquely decodable) The code c : S → T ∗ is uniquely decodable (or UD for short) if the extended function c : S ∗ → T ∗ is an injection. This means that any string in T ∗ corresponds to at most one message in S ∗ . In Chapter 2 we shall explain how the UD property can be guaranteed by a simple construction.

EXERCISES 1.7. A binary code is defined by the rule s1 → 10,

s2 → 010,

s3 → 100.

Show by means of an example that this code is not uniquely decodable. 1.8. Suppose the code c : S → T ∗ is such that every codeword c(s) has the same length n. Is this code uniquely decodable? 1.9. Express the coding rules used in Exercise 1.5 as functions c : S → T ∗ , for suitable alphabets S and T .

1.4 Coding for economy

7

1.4 Coding for economy When the electric telegraph was first introduced, it could transmit only simple electrical impulses. Thus, in order to send messages in a natural language it was necessary to code them into an alphabet with very few symbols. A suitable code was invented by Samuel Morse (1791-1872). The Morse Code uses an alphabet of three symbols: {•, −, ⊙}. The • (dot, pronounced di) is a short impulse, the − (dash, pronounced dah) is a long impulse, and the ⊙ is a pause. Every codeword comprises dots and dashes, ending with a pause. (Strictly speaking, there are also symbols for the shorter pause that separates the dots and dashes within a codeword, and for the longer pause at the end of a message word, but we shall ignore them for the sake of simplicity.) Here are the codewords for A, B, C, D, E, F, X, Y, Z. A →  D → 

•−⊙ − • •⊙

B → E →

X →

− − • • −⊙ Y →

−•••⊙ •⊙

C → F →

−•−•⊙ ••−•⊙

− • − − ⊙ Z →

−−••⊙

In Chapters 2, 3, and 4 we shall look at the basic theory of economical coding and explain how it can be applied to the compression of data. This subject has become very important, because huge amounts of data are now being generated and transmitted electronically.

EXERCISES 1.10. Search the internet to find the standard version of Morse Code, known as the International Morse Code. If this code is defined formally as a function S → T ∗ , what are the alphabets S and T ? 1.11. Decode the following Morse messages: • • • ⊙ − − − ⊙ • • •⊙ ; −−⊙•−⊙−•−−⊙−••⊙•−⊙−•−−⊙ . 1.12. Suppose we try to use a version of Morse code without the symbol ⊙ that indicates the end of each codeword. What is the code for BAD? Find another English word with the same code, showing that this is not a uniquely decodable code.

8

1. Coding and its uses

1.13. The semaphore code enables messages to be exchanged between people who can see each other. Each person has two flags, each of which can be displayed in one of eight possible positions. The two flags cannot occupy the same position. How many symbols can be encoded in this way, remembering that the coding function must be an injection?

1.5 Coding for reliability It is frequently necessary to send messages through unreliable channels, and in such circumstances we should like to use a method of coding that will reduce the likelihood of a mistake. An obvious technique is simply to repeat the message. For example, suppose an investor communicates with a broker by sending the symbols B and S (B = Buy and S = Sell). With this code, if any symbol is received incorrectly, the broker will make a mistake, and perform the wrong action. However, suppose the investor uses the code Buy → BB and Sell → SS. Now if any one symbol is received incorrectly the broker will know that something is wrong, because BS and SB are not codewords, and will be able to ask for further instructions. If the investor uses more repetitions the broker may be able to make a reasonable decision about the intention, even when it is not possible to ask for further instructions. Suppose the investor uses the codewords BBB and SSS. Then, if SSB is received, it is more likely that the message was SSS, because that would imply that only one error had occurred, whereas BBB would imply that two errors had occurred. In Chapters 6-9 we shall describe more efficient methods of coding messages so that the probability of a mistake due to errors in transmission is reduced.

EXERCISES 1.14. Suppose an investor uses the 5-fold repetition code, that is, Buy → BBBBB, Sell → SSSSS. If the following messages are received, which instruction is more likely to have been sent in each case? BBBSB

SBSBS

SSSSB

1.15. Suppose we wish to send the numbers 1, 2, 3, 4, 5, 6, representing the outcomes of a throw of a die, using binary codewords, all of the

1.6 Coding for security

9

same length. What is the smallest possible length of the codewords? Suppose it is required that the receiver will notice whenever one bit in any codeword is in error. Find a set of codewords with length four which has this property.

1.6 Coding for security One of the oldest codes is said to have been used by Julius Caesar over two thousand years ago, with the intention of communicating secretly with his army commanders. For a message in the 27-symbol alphabet A, the rule is: choose a number k between 1 and 25 and replace each letter by the one that is k places later, in alphabetical order. The rule is extended in an obvious way to the letters at the end of the alphabet, as in the example below. The space ⊔ is not changed. Thus if k = 5 the symbols are replaced according to the rule: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ⊔ F G H I J K L M N O P Q R S T U V W X Y Z A B C D E ⊔

.

becomes

.

For example, the message

SEE⊔YOU⊔TOMORROW

XJJ⊔DTZ⊔YTRTWWTB

In mathematical terms the coding rule is a function ck : A → A, which depends on the key k: in the example given above, k = 5. It is a basic assumption of cryptography that, although the value of k may be kept secret, the general form of the coding rule cannot. In other words, it will become known that the rule is ck (apply a shift of k to the letters) for some k. When a coded message such as XJJ⊔DTZ⊔YTRTWWTB is sent, it is presumed that the intended recipient knows the key – the value k = 5 in our example. In that case it is easy to decode the message. On the other hand, if someone who does not know the key intercepts the message, decoding is not necessarily so easy. In cryptography, decoding by finding the value of the key k (or otherwise) is said to be breaking the system, and any method which may achieve this is an attack. In fact, Caesar’s system is not very secure, because there is a simple attack by the method known as exhaustive search. The only possible values of k are

10

1. Coding and its uses

1, 2, 3, . . . , 25, and it is easy to try each of them in turn, until a meaningful message is found.

Example 1.12 Suppose we have intercepted the message SGZNY⊔OY⊔MUUJ⊔LUX⊔EUA

.

We suspect that Caesar’s system is being used. How do we find the key? Solution Trying the possible keys, beginning with k = 1 and k = 2 produces the following possibilities. Remember that if the key is k, we must go back k places to find the original message. k=1: k=2:

RFYMX⊔NX⊔ QEXLW⊔MW⊔

··· ···

Thus the key is not 1 or 2, because if it were, the original message would not make sense. There is no need to ‘decode’ the whole message in order to establish this fact. So we must continue to work through the keys k = 3, 4, . . . , 25, until a meaningful message is found.

EXERCISES 1.16. Find the original message in Example 1.12. 1.17. Could the following message have been sent by Julius Caesar himself? ZLJB⊔LK⊔BKDIXKA 1.18. Caesar’s system is an example of a substitution code, because each letter in the message is replaced by a substitute letter, according to a fixed rule. Suggest other substitution rules, with a view to defending against the attack by exhaustive search.

Further reading for Chapter 1 The internet is a treasury of information about Morse code, semaphore, and other historically important coding systems. The pioneering work of Claude Shannon on the theory of information and communication is also wellrepresented.

1.6 Coding for security

11

Internet sites relating to cryptography are very variable in quality, and it is better to rely on good books such as those by Kahn [1.2] and Singh [1.3]. Older books on cryptography can also provide an important perspective for understanding the modern approach. The books by d’Agapeyeff [1.1] and Sacco [1.4] are recommended. Books about the so-called ‘Bible Codes’ and similar matters should be regarded as entertainment. They are more entertaining (often unintentionally) when considered from the viewpoint of an informed reader, such as someone who has studied this book. 1.1 A. d’Agapeyeff. Codes and Ciphers. Oxford University Press, London (1939). 1.2 D. Kahn. The Codebreakers. Scribner, New York (1996). 1.3 S. Singh. The Code Book. Fourth Estate, London (2000). 1.4 L. Sacco. Manuel de Cryptographie. Payot, Paris (1951).

2 Prefix-free codes

2.1 The decoding problem The symbols in a message appear in a certain order. So we often think of a message as part of a stream ξ1 ξ2 ξ3 · · · , with the order being determined by a process that takes place in real time. Here each ξk is a variable that can take as its value any symbol from the alphabet S, and its actual value is the symbol that occurs at time k (k = 1, 2, 3, . . .). In this chapter we study the basic facts about coding and decoding a stream of symbols. We shall say that a coding function c : S → T ∗ replaces the original stream of symbols belonging to the alphabet S by an encoded stream of symbols belonging to the alphabet T .

Example 2.1 Let S = {x, y, z} and suppose the original stream is yzxxzyxzyy · · ·

.

The code c : S → B∗ is defined by the rules x → 0,

y → 10,

z → 11.

What is the encoded stream? N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 2, 

14

2. Prefix-free codes

Solution The encoded stream is simply the concatenation of the codewords for the individual symbols: 10110011100111010 · · ·

.

It is reasonable to require that the code c is uniquely decodable (Definition 1.11), so that the encoded stream corresponds to a unique original stream. Better still, we should like to decode the encoded stream in a sequential fashion, without having to wait for the message to be complete before we begin. This corresponds (very roughly) to what we do when we read a written message: we decode word-by-word and sentence-by-sentence. In that situation the decoding process is made easier by the presence of spaces (indicating end-of-word) and full stops (indicating end-of-sentence). It must be stressed that, in general, such helpful indications will not be available. Here is an example in which sequential decoding is possible.

Example 2.2 Suppose we receive the encoded stream 100011100 · · ·

,

and we are provided with a ‘codebook’ that tells us that the coding rule is as in Example 2.1. That is, x → 0,

y → 10,

z → 11.

How do we find the original stream? Solution The first symbol 1 is not in the codebook, so we look at the next symbol, 0. The string 10 is in the codebook as the code for y, so we decide that the first symbol of the original stream is y. The next symbol 0 is the codeword for x, so we decide that the next symbol is x. Continuing in this way we obtain the original stream yxxzy · · · . In general, for a code c : S → T ∗ the method used in Example 2.2 is as follows. Examine the symbols in the encoded stream in order, until a string q that is a complete codeword is recognized. Since c is an injection (by definition), there is a unique symbol s ∈ S such that c(s) = q. So we decode this part of the stream as s, delete it, and repeat the process. Clearly, this method will fail if the codeword q is also the prefix of another codeword q ′ : that is, if q ′ = qr for some nonempty word r ∈ T ∗ . In that case, we cannot distinguish between the complete word q and the initial part of q ′ .

2.1 The decoding problem

15

Definition 2.3 (Prefix-free) We say that a code c : S → T ∗ is prefix-free (or simply PF) if there is no pair of codewords q = c(s), q ′ = c(s′ ) such that q ′ = qr,

for some nonempty word r ∈ T ∗ .

Example 2.4 Let S = {w, x, y, z} and define c : S → B∗ by w → 10,

x → 01,

y → 11,

z → 011.

Is this code PF, and is it uniquely decodable? Solution It is clearly not PF, since c(x) = 01 is a prefix of c(z) = 011. In fact it is not uniquely decodable either. For example, the string 10011011 corresponds to two different messages, wxwy and wzz: wxwy → 10 01 10 11

wzz → 10 011 011.

(The extra spaces are shown only to clarify the explanation.) This example shows that a code that is not PF need not be UD. However, it is almost obvious that any PF code is UD. The formal proof is based on the sequential decoding method used in Example 2.2.

Theorem 2.5 If a code c : S → T ∗ is prefix-free, then it is uniquely decodable.

Proof Suppose that x1 x2 · · · xm and y1 y2 · · · yn are strings in S ∗ such that their codes c(x1 x2 · · · xm ) and c(y1 y2 · · · yn ) are the same. That is c(x1 )c(x2 ) · · · c(xm ) = c(y1 )c(y2 ) · · · c(yn ). Since these strings are the same, their initial parts are the same. It follows that if c(x1 ) = c(y1 ) then one is a prefix of the other, contrary to hypothesis. Hence c(x1 ) = c(y1 ), and since c is an injection, x1 = y1 . Thus the remaining parts of the two strings are the same. Repeating the same argument, it follows that x2 = y2 , and so on. Hence x1 x2 · · · xm = y1 y2 · · · yn (and m = n), so c is UD.

16

2. Prefix-free codes

At first sight, it would seem that PF codes are very special. There are many possible decoding rules, and a code that is UD need not be PF. (See Exercise 2.3.) However, there are good reasons why PF codes are the only ones we need consider seriously. In the rest of this chapter we shall explain why this is so.

EXERCISES 2.1. A prefix-free binary code is defined by s1 → 00, s2 → 010, s3 → 100, s4 → 111. If the encoded stream is 1001110100011101010000, what is the original stream? 2.2. In Exercise 1.7 we found that the binary code s1 → 10,

s2 → 010,

s3 → 100

is not uniquely decodable. What is the underlying reason for this? 2.3. In the previous exercise, replace the coding of s2 by s2 → 1. Show that although the new code is not prefix-free, it is uniquely decodable. 2.4. Explain carefully why the Morse code, as defined in Section 1.4, is prefix-free. Explain also why the modified version (without ⊙) discussed in Exercise 1.12 is not prefix-free.

2.2 Representing codes by trees There is a useful way of representing a code by means of a tree. It works for codes in an alphabet of any size b, but for the purposes of illustration we shall focus on the binary alphabet B with b = 2. First we observe that the set of nodes of an infinite binary tree (see Figure 2.1) can be regarded as B∗ , the set of all words in B. Precisely, the root of the tree is labelled by the empty word, and the other nodes are labelled recursively, so that the two ‘children’ of the node labelled w are labelled w0 and w1. We shall follow the standard (but perverse) practice of using a mixture of botanical and genealogical terms to describe a tree. For example, these trees ‘grow’ downwards, with the root at the top ‘level’.

2.2 Representing codes by trees

17

0

00

000

1

01

001

010

10

011

100

11

101

110

111

Figure 2.1 The nodes of the infinite binary tree correspond to B∗ Now suppose that a code C ⊆ B∗ is given, and the codewords are represented by the corresponding nodes of the binary tree. Clearly, a codeword q is a prefix of another codeword q ′ if the unique path from q ′ to the root of the tree passes through q. If the code C is prefix-free it follows that, for any codeword q, none of the descendants of q can be a codeword. Thus we can ignore all the nodes that are descendants of q. If we also ignore all those nodes that are neither codewords nor prefixes of codewords, we have a finite binary tree, and the codewords of C are its leaves.

Example 2.6 Represent the code C = {0, 10, 110, 111} by means of a tree. Solution

The tree is shown in Figure 2.2.

0 10 110

111

Figure 2.2 A set of codewords represented as the leaves on a tree

18

2. Prefix-free codes

EXERCISES 2.5. Sketch the tree that represents the PF code s1 → 00, s2 → 010, s3 → 100, s4 → 111, and label each leaf with the corresponding symbol. Is it possible to extend this code without destroying the PF property? 2.6. Denote the ternary alphabet by T = {0, 1, 2}. Construct a tree representing the ternary code with codewords 02, 101, 120, 221, 222. Is it possible to extend this code without destroying the PF property? 2.7. What is the maximum number of codewords in a prefix-free binary code in which the longest codeword has length 7? 2.8. In Exercise 1.15 we discussed the problem of representing the numbers 1, 2, 3, 4, 5, 6 by a binary code, in such a way all six codewords have the same length. Suppose we now ask for a PF binary code, such that the total length of the codewords is as small as possible. Construct a suitable code, using the tree representation as a guide.

2.3 The Kraft-McMillan number Given a code c : S → T ∗ , let ni denote the number of symbols in S that are encoded by strings of length i in T ∗ . In other words, ni is the number of s ∈ S such that c(s) is in T i . If M is the maximum length of a codeword, we refer to the numbers n1 , n2 , . . . , nM as the parameters of c. In Figure 2.2 the parameters are n1 = 1, n2 = 1, n3 = 2. In general, in the tree diagram, ni is the number of codewords at level i. When |T | = b the total number of strings of length i in T ∗ is |T i | = bi . By definition, a code c : S → T ∗ is an injection, so its parameters must satisfy ni ≤ bi . The fraction ni /bi represents the proportion of words of length i that are used as codewords.

Definition 2.7 (Kraft-McMillan number) The Kraft-McMillan number associated with the parameters n1 , n2 , . . . , nM , and the base b, is defined to be K=

M  ni i=1

bi

=

n2 n1 nM + 2 + ··· + M . b b b

2.3 The Kraft-McMillan number

19

For example, if n1 = 1, n2 = 2, n3 = 1, and b = 2, then K=

9 1 2 1 + + = . 2 4 8 8

We shall prove two important results about the existence of b-ary codes. • If K ≤ 1 for a set of parameters n1 , n2 , . . . , nM then there is a PF b-ary code with those parameters. • The parameters of a UD b-ary code must satisfy the condition K ≤ 1. Taken together, these two results imply that if there is a UD code with certain parameters, then there is PF code with the same parameters. The proof of the first result is best explained in terms of the tree representation. The basic idea is illustrated in the following example.

Example 2.8 Construct a PF binary code with parameters n2 = 2, n3 = 3, n4 = 1. Solution

First, we check that the K ≤ 1 condition is satisfied: K=

1 15 2 3 + + = ≤ 1. 4 8 16 16

For the construction, since n2 = 2 we start with two codewords of length 2, say 00 and 01. The PF condition means that we cannot use any words of length 3 the form 0 0∗ or 0 1∗, but there remain four other possible words, of the form 1 ∗ ∗. In fact we require only three of them, so we can choose 100, 101, 110, for example. Finally, we require one word of length 4, which can be either 1110 or 1111.

Theorem 2.9 If K ≤ 1 for the parameters n1 , n2 , . . . , nM , then a PF b-ary code with these parameters exists.

Proof First, note that in the inequality K=

n2 n1 nM + 2 + · · · + M ≤ 1, b b b

each term ni /bi is non-negative. This means that any partial sum of these terms is also not greater than 1:

20

2. Prefix-free codes

n1 n2 n2 n1 n1 n3 ≤ 1, + 2 ≤ 1, + 2 + 3 ≤ 1, b b b b b b These inequalities can be rewritten as

··· .

n1 ≤ b,

n2 ≤ b(b − n1 ),

n3 ≤ b(b2 − n1 b − n2 ),

··· ,

and generally, ni ≤ b(bi−1 − n1 bi−2 − · · · − ni−1 ). Since n1 ≤ b, it is possible to choose n1 different codewords of length 1. There remain b − n1 words of length 1, each of which gives rise to b words of length 2, and the PF condition means that only these b(b − n1) words of length 2 are available as codewords. Since n2 ≤ b(b − n1 ), it is possible to choose n2 of them as codewords (Figure 2.3).

n1

b − n1 n2

b(b − n1 ) − n2

Figure 2.3 Finding codewords when K ≤ 1: initial steps As before, the PF condition means that only b(b2 − n1 b − n2 ) words of length 3 are available as codewords. Since n3 ≤ b(b2 − n1 b − n2 ), it is possible to choose n3 of them. The general step is illustrated in Figure 2.4. The number of unused codewords of length i − 1 is fi−1 = bi−1 − n1 bi−2 − · · · − ni−1 and since ni ≤ bfi−1 , it is possible to choose ni words of length i as codewords without violating the PF condition. Thus the condition K ≤ 1 ensures that enough codewords are available at each step, and a PF code with the given parameters can be constructed.

2.4 Unique decodability implies K ≤ 1

21

fi−1

ni

bfi−1 − ni

Figure 2.4 Finding codewords when K ≤ 1: general step

EXERCISES 2.9. Construct PF binary codes with the following parameters: (i) n2 = 1, n3 = 4, n4 = 3; (ii) n1 = 1, n2 = 0, n3 = 2, n4 = 3, n5 = 2. 2.10. On the basis of Theorem 2.9, what can be said about the existence of prefix-free ternary (b = 3) codes with the following parameters? (i) n1 = 0, n2 = 1, n3 = 12; (ii) n1 = 0, n2 = 1, n3 = 12, n4 = 40. 2.11. If the conclusion of the previous exercise is that a prefix-free code must exist, construct one. 2.12. Let us say that a code is complete if it is PF and K = 1. Show that if there is a complete b-ary code for an alphabet of size m, then b − 1 must divide m − 1.

2.4 Unique decodability implies K ≤ 1 In this section we prove that the parameters of a UD code must satisfy K ≤ 1. The proof involves some elementary combinatorial theory. Let c : S → T ∗ be a code with parameters n1 , n2 , . . . , nM . Since M is the length of the longest codeword for a single symbol, the maximum length of the code for a string of r symbols is rM .

22

2. Prefix-free codes

Given r ≥ 1, let qr (i) be the number of strings of length r that are encoded by strings of length i (1 ≤ i ≤ rM ); in particular, q1 (i) = ni . We shall show that the parameters ni determine the numbers qr (i), for all r.

Example 2.10 Suppose c : S → T ∗ is a code with parameters n1 = 2, n2 = 3. What are the numbers q2 (i)? Solution Let xy be a string of length 2 in S. Since c(x) and c(y) have length 1 or 2, c(xy) has length 2, 3, or 4. If c(xy) has length 2, c(x) and c(y) must both have length 1. Since n1 = 2, there are 2 × 2 possible strings xy. Thus q2 (2) = 4. If c(xy) has length 3, c(x) must have length 1 or 2, and c(y) must have length 2 or 1, respectively. Since n1 = 2 and n2 = 3, there are 2 × 3 + 3 × 2 possible strings xy. Thus q2 (3) = 6 + 6 = 12. If c(xy) has length 4, c(x) and c(y) must both have length 2. Since n2 = 3, there there are 3 × 3 possible strings xy. Thus q2 (4) = 9. We define the generating function for the sequence qr (1), qr (2), . . . qr (rM ), as follows: Qr (x) = qr (1)x + qr (2)x2 + · · · + qr (rM )xrM .

Example 2.11 What is the relationship between Q1 (x) and Q2 (x) for the code described in the previous example? Solution

The generating functions for r = 1 and r = 2 are Q1 (x) = 2x + 3x2 ,

Q2 (x) = 4x2 + 12x3 + 9x4 ,

so that Q2 (x) = Q1 (x)2 . This example suggests the following result, which we refer to as the Counting Principle. The proof will be given in Section 2.5.

Lemma 2.12 For all r ≥ 1, Qr (x) = Q1 (x)r . Assuming this result, we can prove the second important theorem about the Kraft-McMillan number.

2.4 Unique decodability implies K ≤ 1

23

Theorem 2.13 If a uniquely decodable b-ary code exists, with a given set of parameters, then its Kraft-McMillan number satisfies K ≤ 1.

Proof Fix r ≥ 1. Putting x = 1/b in the definition of Qr (x) we have Qr (1/b) = qr (1)/b + qr (2)/b2 + · · · + qr (rM )/brM . Unique decodability implies that qr (i) cannot exceed bi , the total number of strings of length i. It follows that every term in Qr (1/b) is less than or equal to 1, and since the number of terms is at most rm, Qr (1/b) < rM . Now the Kraft-McMillan number is given by K=

n2 nM n1 + 2 + · · · + M = Q1 (1/b). b b b

Thus, by the Counting Principle K r = Q1 (1/b)r = Qr (1/b). Since Qr (1/b) < rM it follows that K r /r < M for all r, where M is a fixed constant. Suppose it were true that K > 1, so that K = 1 + h with h > 0. By the binomial theorem, 1 K r = (1 + h)r = 1 + rh + r(r − 1)h2 + · · · . 2 Since all terms are positive, K r is certainly greater than the term in h2 , that is 1 Kr > (r − 1)h2 . r 2 Thus by taking r sufficiently large, we could make K r /r as large as we please, contrary to the result obtained above. So we must have K ≤ 1. Theorem 2.13 can be combined with Theorem 2.9 as follows: UD code exists

=⇒

K≤1

=⇒

PF code exists.

The combined result is that the existence of a UD code with given parameters implies the existence of a PF code with the same parameters. We can therefore say that ‘PF codes suffice’. The converse of this result was proved in Theorem 2.5. So we conclude that there is a UD code with certain parameters if and only if there is a PF code with the same parameters, and this happens if and only if K ≤ 1.

24

2. Prefix-free codes

EXERCISES 2.13. Suppose we wish to construct a UD code for 12 symbols using binary words of length not exceeding 4. Make a list of all the sets of parameters n1 , n2 , n3 , n4 for which a suitable code exists. 2.14. Let S = {s1 , s2 , . . . , s6 } and T = {a, b, c}, and suppose that a code is defined by s1 → a, s2 → ba, s3 → bb, s4 → bc, s5 → ca, s6 → cb. Write down the generating function Q1 (x) and hence (by algebraic means) find Q2 (x). 2.15. In the previous exercise, what does the coefficient of x4 in Q2 (x) represent? Verify your answer by making a list of the corresponding elements of S ∗ . 2.16. Let S = {a, b, c, d, e, f, g} and suppose that a binary code is defined by a → 00, b → 010, c → 011, d → 1000, e → 1001, f → 1101, g → 1111. Write down the generating function Q1 (x) and hence calculate Q2 (x) and Q3 (x). What do the coefficients of x7 in Q2 (x) and Q3 (x) represent? Verify your answers by making lists of the corresponding elements of S ∗ .

2.5 Proof of the Counting Principle Theorem 2.14 (The Counting Principle) Let c : S → T ∗ be a code such that, for all s ∈ S, the length of c(s) is not greater than M . Given r ≥ 1, let qr (i) be the number of strings of length r in S that are encoded by strings of length i in T (1 ≤ i ≤ rM ). Let Qr (x) be the generating function Qr (x) = qr (1)x + qr (2)x2 + · · · + qr (rM )xrM . Then Qr (x) = Q1 (x)r .

2.5 Proof of the Counting Principle

25

Proof We consider the coefficient of xN in the product Q1 (x)r = (n1 x + n2 x2 + · · · + nM xM )r , where, as usual, ni = q1 (i). A contribution to the coefficient of xN is obtained by multiplying r terms, one from each of the r factors. That is, we choose a term ni1 xi1 from the first factor, ni2 xi2 from the second factor, and so on, in such a way that i1 + i2 + · · · + ir = N , so that the resulting power of x is N . Hence the coefficient of xN is the sum of all products ni1 ni2 . . . nir

where i1 + i2 + · · · + ir = N.

Now consider qr (N ), the number of strings of length N that can be formed from the strings representing r symbols in S. Such a string is formed by choosing any one of ni1 codewords of length i1 , ni2 codewords of length i2 , and so on, in such a way that i1 + i2 + · · · + ir = N . Hence qr (N ) is just the coefficient of xN , as in the previous paragraph.

EXERCISES 2.17. For a given code c : S → T ∗ , let C(x, y) be the two-variable generating function for the numbers qr (i) defined above, that is  qr (i)xi y r . C(x, y) = i,r

Prove that C(x, y) = yQ1 (x)/(1 − yQ1 (x)).

Further reading for Chapter 2 The number that we have denoted by K, and called the ‘Kraft-McMillan’ number, has an interesting history. Our Theorem 2.9 (K ≤ 1 implies that a PF code exists) was proved by L.G. Kraft in his 1949 Master’s thesis [2.3]. He also gave a proof of the converse result. The converse also follows from our Theorem 2.13 (UD implies that K ≤ 1), which was first proved by B. McMillan in 1956 [2.4] using a method based on complex variable theory. A simpler proof was discovered by J. Karush in 1961 [2.2], and our proof is based on Karush’s method.

26

2. Prefix-free codes

The fact that it is an application of a standard result on generating functions is often overlooked. A good account of the early history of this topic is available in the collection of articles edited by E.R. Berlekamp [2.1]. 2.1 E.R. Berlekamp. Key Papers in the Development of Coding Theory. IEEE Press, New York (1974). 2.2 J. Karush. A simple proof of an inequality of McMillan. IRE Trans. Information Theory IT-7 (1961) 118. 2.3 L.G. Kraft. A device for quantizing grouping, and coding amplitude modulated pulses, M.S. thesis, Electrical Engineering Department, MIT (1949). 2.4 B. McMillan. Two inequalities implied by unique decipherability. IRE Trans. Information Theory IT-2 (1956) 115-116.

3 Economical coding

3.1 The concept of a source Roughly speaking, a ‘source’ is a means of producing messages. Examples are a human being writing email messages, or a scanner making digitized images. Using the terminology introduced in Chapter 2, we say that a source emits a stream of symbols, denoted by . ξ1 ξ2 ξ3 · · · Here each ξk is a variable that can take as its value any symbol belonging to a given alphabet. An obvious feature that distinguishes one source from another is the alphabet. Writers in oriental languages use alphabets that are quite different from the 27-letter alphabet that we have denoted by A. However, there are other distinctive features. Writers in western languages all use alphabets similar to A, but in different ways. In a natural language there are certain rules about how the alphabet is used (spelling, grammar, syntax) and these are reflected in the stream emitted by the source. For example, in English the symbol J will occur much less often than the symbol E. Similarly, if a scanner is set up to look at text documents, it will produce messages in which W (representing a white pixel) is much more likely than B (representing a black pixel). These remarks lead to the conclusion that a source can be described by specifying both the alphabet and the probabilities of the symbols in the stream emitted. Suppose, for example, that the source is an experiment in which a coin is tossed repeatedly, the results being recorded as H = Head, T = T ail. N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 3, 

28

3. Economical coding

If the coin is a fair one, the stream emitted might be HT T HT HT HT T HHT T T HT HHT T HT HT HT H . . . , where H and T occur with roughly the same frequency. This is a source with alphabet S = {H, T }. The probability that ξk is H, and the probability that ξk is T , are both 12 . That is, for all k, Pr(ξk = H) = Pr(ξk = T ) = 0.5. On the other hand, an unfair coin would emit a stream such as HHHHHHHT HHHHHT HHHHHHHHHHT . . . , where the probabilities are such that Pr(ξk = H) is significantly greater than Pr(ξk = T ). The theory required to set up a complete mathematical model of a source is, in general, fairly complicated, but we can make some progress by looking at a very simple model.

Definition 3.1 (Probability distribution) Let S = {s1 , s2 , . . . , sm } be an alphabet. A probability distribution on S is a set of real numbers p1 , p2 , . . . , pm such that p1 + p2 + . . . + pm = 1,

0 ≤ pi ≤ 1 (i = 1, 2, . . . , m).

We shall usually write the numbers pi as a row vector p = [p1 , p2 , . . . , pm ], and say that the symbol si occurs with probability pi . Given a probability distribution p on S, consider a source with the following property. The source emits a stream ξ1 ξ2 ξ3 · · ·, where the value of each ξk is a member of the alphabet S and, for all k, the probability that ξk takes a particular value si is given by Pr(ξk = si ) = pi . Technically, ξk is a random variable. We shall refer to this model as a ‘source with probability distribution p’ or simply a ‘source (S, p)’. It is important to stress that the terminology does not imply that all the characteristics of the source are described by p, it simply means that the probability distribution for each individual ξk is the same. In many cases this is a reasonable assumption, even though the source may have other, more subtle, features. For example, suppose we regard the text of a book as a stream of symbols in the alphabet A. If we open the book at page x and find the yth symbol on that page, then the probability that the symbol

3.1 The concept of a source

29

is E can reasonably be assumed to be the same, for all x and y. But if we know that the 17th letter on page 83 is Q, then that affects the probability that the 18th letter on page 83 is E. In technical terms, we are imposing the condition that the random variables ξk all have the same distribution p but, in general, we do not assume that these random variables are independent.

Definition 3.2 (Memoryless source) A source (S, p) that emits a stream ξ1 ξ2 ξ3 · · · is memoryless if the random variables ξk are independent. That is, for all k and ℓ Pr(ξk = si and ξℓ = sj ) = Pr(ξk = si ) Pr(ξℓ = sj ). For a memoryless source, the distribution p provides a complete description: knowledge of any of the terms in a message does not affect the probability distributions assigned to the other terms. This property can be assumed to hold in the coin-tossing experiment described above, and in some other real situations.

Example 3.3 In the UK there is a weekly ritual called the ‘football results’. This can be regarded as a source that emits a stream of symbols chosen from the alphabet S = {h, a, d}, where h = home win, a = away win, d = draw. Is this a memoryless source? Solution It is reasonable to assume that this source is memoryless: the result of one football match does not affect the result of any other matches in the list. Observation suggests that the probability distribution is, roughly, ph = 0.42,

pa = 0.26,

pd = 0.31.

Again, it must be stressed that many real sources are not memoryless. We shall return to this point in the next chapter. Even in the memoryless case, there are questions about how probabilities should be interpreted in real situations. The coin-tossing example indicates that it may be necessary to infer the characteristics of the source by examining the stream that it emits. The purpose of the experiment might be to check whether or not the coin is a fair one. We shall generally assume that the probability of an event represents the relative frequency of its occurrence, but for some purposes it is helpful to adopt an alternative viewpoint in which the probability is regarded as a ‘degree of belief’ (the Bayesian approach).

30

3. Economical coding

EXERCISES 3.1. Consider a source emitting symbols from the alphabet S with probability distribution p, where S = {a, b, c}

p = [pa , pb , pc ] = [0.6, 0.3, 0.1].

If this source emits a stream of 100 symbols, approximately how many times will a occur? If the source is memoryless, approximately how may times will the consecutive pair ab occur? 3.2. An instrument records the temperature in degrees Celsius every hour at a certain place. Is this a memoryless source? 3.3. A source emits the stream of positive integers ξ1 ξ2 ξ3 · · · defined by the rules ξ1 = 1, ξ2 = 1, ξk = ξk−1 + ξk−2 (k ≥ 3). How does this source differ from the ones discussed in this section?

3.2 The optimization problem We now consider what happens when the original stream emitted by a source (S, p) is transformed into an encoded stream, using an alphabet T . In particular, how does the length of a typical message depend on the code c : S → T ∗ that is used? Suppose that S = {s1 , s2 , . . . , sm }. Let yi be the length of the codeword c(si ) (1 ≤ i ≤ m), and consider a message of length N in S. If N is a reasonably large number, the message contains the symbol s1 approximately N p1 times, s2 approximately N p2 times, and so on. After encoding with c, there are N p1 strings c(s1 ), each of length y1 , N p2 strings c(s2 ), each of length y2 , and so on. The total length of the encoded message is approximately N p1 y1 + N p2 y2 + · · · N pm ym = N (p1 y1 + p2 y2 + · · · + pm ym ). Thus if the original message has length N , the encoded message has length LN , approximately, where L = p1 y1 + p2 y2 + · · · + pm ym .

Definition 3.4 (Average word-length) The average word-length of a code c : S → T ∗ for the source (S, p) is L = p1 y 1 + p2 y 2 + · · · + pm y m .

3.2 The optimization problem

31

Example 3.5 Let S = {s1 , s2 , s3 } and p = [0.2, 0.6, 0.2]. Find the value of L when the binary code s1 → 0, s2 → 10, s3 → 11 is used. Is there a binary code for S that achieves a smaller value? Solution

The value of L is 1 × 0.2 + 2 × 0.6 + 2 × 0.2 = 1.8.

Noting that s2 is the most likely symbol, we should try a code that uses the shortest codeword for this symbol, such as s1 → 10,

s2 → 0,

s3 → 11.

Here the value of L is 2 × 0.2 + 1 × 0.6 + 2 × 0.2 = 1.4. In this example it is fairly easy to improve on the original code, but we did not ask whether further improvements are possible. Question: Given the source (S, p) and the alphabet T , how can we find a code c : S → T ∗ for which L is as small as possible? For practical purposes, it is important that the code is uniquely decodable, so we make the following definition.

Definition 3.6 (Optimal code) Given a source (S, p) and an alphabet T , a UD code c : S → T ∗ is optimal if there is no such code with smaller average word-length. The requirement that the code be UD is a significant constraint. In Chapter 2 we showed that this constraint can be expressed by the condition K ≤ 1, where K is the Kraft-McMillan number associated with parameters n1 , n2 , . . . , nM and the base b of the code. Explicitly K=

n2 n1 nM + 2 + ···+ M . b b b

In our current notation, ni is the number of symbols sj such that yj = i, and M is the maximum value of yj . The term ni /bi in K is the sum of ni terms

32

3. Economical coding

1/bi , corresponding to one term 1/byj for each j such that yj = i. It follows that K can be written as the sum of all the terms 1/byj : K=

1 1 1 + y2 + · · · + ym . y 1 b b b

Rewriting the K ≤ 1 condition in this form we can formulate the problem of finding optimal codes as follows. Given b and p1 , p2 , . . . , pm find positive integers y1 , y2 , . . . , ym that minimize

p1 y 1 + p2 y 2 + · · · + pm y m

1 1 1 + y2 + · · · + ym ≤ 1. y 1 b b b This problem cannot be solved completely by ‘calculus methods’, due to the condition that the yi ’s must be integers. In the following sections we shall discuss it using an approach based on one of the most important concepts in coding theory. subject to

EXERCISES 3.4. A source emits three symbols with probabilities 0.5, 0.25, 0.25. Construct a PF binary code for this source with average word-length 1.5. 3.5. Consider the general case of a source emitting three symbols with probability distribution [α, β, 1 − α − β]

where α > β > 1 − α − β ≥ 0.

Show that the average word-length of an optimal binary code for this source is 2 − α.

3.3 Entropy Definition 3.7 (Entropy of a distribution) The entropy to base b of a probability distribution p = [p1 , p2 , . . . , pm ] is Hb (p) = Hb (p1 , p2 , . . . , pm ) =

m  i=1

pi logb (1/pi ).

3.3 Entropy

33

If 0 < pi < 1 then 1/pi > 1, and each term pi logb (1/pi ) is positive. Occasionally it is convenient to allow pi = 0, when the expression pi logb (1/pi ) is strictly not defined. However, we shall give it the conventional value 0, since that is its limit as pi → 0. Since the output of a memoryless source (S, p) is completely determined by p, we shall often speak of the entropy of p as the entropy of the source. However, it must be stressed that many sources are not memoryless, and for them a more sophisticated definition of entropy is required (see Chapter 4).

Example 3.8 What is the entropy to base 2 of a memoryless source with distribution p = [0.5, 0.25, 0.25]? Solution H2 (p1 , p2 , p3 )

= p1 log2 (1/p1 ) + p2 log2 (1/p2 ) + p3 log2 (1/p3 ) = 0.5 log2 (1/0.5) + 0.25 log2 (1/0.25) + 0.25 log2 (1/0.25) = 0.5 log2 2 + 0.25 log2 4 + 0.25 log2 4 = 0.5 × 1 + 0.25 × 2 + 0.25 × 2 = 1.5.

EXERCISES 3.6. What is the entropy to base 2 of a memoryless source emitting five letters A,E,I,O,U with probabilities 0.2, 0.3, 0.2, 0.2, 0.1? What is the entropy if all five letters are equally probable? 3.7. What is the entropy of a source that is certain to emit one specific symbol? 3.8. What is the entropy of a memoryless source that emits symbols from an alphabet of size m, each symbol being equally probable? 3.9. Suppose that m men and w women enter a ‘reality TV’ contest. The probability that the ith man will win is ui and the probability that the jth woman will win is vj . Let U = u1 + u2 + . . . um ,

V = v1 + v2 + . . . + vw .

34

3. Economical coding

Show that the distribution [u1 , u2 , . . . , um , v1 , v2 , . . . , vw ] has entropy equal to H(U, V ) + U H(u1 /U, u2 /U, . . . , um /U ) + V H(v1 /V, v2 /V, . . . vw /V ). (This result has a simple interpretation: see Exercise 3.12.)

3.4 Entropy, uncertainty, and information At this point the definition of entropy is unmotivated. What is its significance? What is the role of the base b? How is entropy relevant to the optimal coding problem? In order to answer these questions, consider first a very simple example, a memoryless source emitting the two symbols 0 and 1. The probabilities p0 and p1 can be written as x and 1 − x for some x in the range 0 ≤ x ≤ 1, and the entropy of this source (to base 2) is h(x) = x log2 (1/x) + (1 − x) log2 (1/(1 − x)). This function turns up frequently, and we shall reserve the letter h for it. The graph of h for 0 ≤ x ≤ 1 is shown in Figure 3.1. It is symmetrical about the line x = 12 and the value h( 21 ) = 1 is the maximum value (Exercise 3.11). h(x) 1

0

1 2

1

x

Figure 3.1 The graph of h(x) Figure 3.1 suggests that entropy is a measure of uncertainty. We would expect that the greatest uncertainty about which symbol is emitted will occcur when the two symbols are equally probable (x = 12 ); on the other hand there is no uncertainty if one of the symbols is never emitted (x = 0 or x = 1). In fact, it can be shown quite generally that the entropy  pi logb (1/pi ), Hb (p) =

3.4 Entropy, uncertainty, and information

35

is a measure of the uncertainty about the identity of a symbol that is taken from a set according to the probability distribution p. To be precise, Hb (p) is essentially the only function that satisfies some very reasonable conditions that we associate with the notion of uncertainty. The word ‘essentially’ is needed here because we can always change the unit of measurement, which corresponds to multiplying by a constant. In fact, this is already part of our definition, since the base b is not specified. If we chose another base a then the identity loga x = loga b × logb x implies that Ha (p) = loga b × Hb (p). Thus changing the base amounts to changing the unit of measurement. The base b = 2 is usually taken as the standard, in which case we write H(p) instead of H2 (p), and we say that H(p) measures the uncertainty in bits per symbol. If it is appropriate to use a different base (as in the next Section) we can use the relation H(p) . Hb (p) = log2 b The concept of information is closely related to uncertainty. Roughly speaking, providing information about an event reduces our uncertainty about it. We can reconcile this idea with our definition of the entropy-uncertainty of p in the following way. Suppose a memoryless source emits a steam of symbols ξ1 ξ2 ξ3 · · ·, where each ξk has the distribution p. The entropy is the sum of terms pi log(1/pi ), each of which is the product of the probability pi that ξk has the value si and the quantity log(1/pi ). If we interpret log(1/pi ) as the amount of information  provided by knowing that ξk = si , then the sum pi log(1/pi ) is, on average, the amount of information provided by knowing the value of ξk . For example, when the source emits two symbols, with probabilities x and 1 − x, the information provided is h(x) bits per symbol. As we noted above, this is greatest when the symbols are equally probable. It is zero when one of the symbols has probability 0, because then we know the value of each ξk in advance. We can now develop some of the mathematical properties of entropyuncertainty-information. A very useful result (Theorem 3.10) relies on the following lemma from elementary calculus.

36

3. Economical coding

Lemma 3.9 For all x > 0, ln x ≤ x − 1, with equality if and only if x = 1. (Figure 3.2.)

Figure 3.2 Graphs of x − 1 and ln x

Proof Let f (x) = x − 1 − ln x. Then f ′ (x) = 1 − 1/x, which is zero only when x = 1. Since f ′′ (1) > 0, the value f (1) = 0 is the minimum value of f , and the result follows.

Theorem 3.10 (The comparison theorem) If p1 , p2 , . . . , pm and q1 , q2 , . . . , qm are probability distributions then Hb (p) =

m  i=1

pi logb (1/pi ) ≤

m 

pi logb (1/qi ).

i=1

There is equality if and only if qi = pi for all i (1 ≤ i ≤ m).

Proof It is sufficient to prove the result for any one value of b, since changing b to b′ amounts to multiplying both sides of the inequality by logb b′ . In fact we shall take b = e, the base of natural logarithms, so that loge x = ln x. By Lemma 3.9, ln(qi /pi ) ≤ (qi /pi ) − 1,

3.4 Entropy, uncertainty, and information

37

with equality if and only if qi = pi . Since ln(qi /pi ) = ln(1/pi ) − ln(1/qi ), we have m m m    pi ln(qi /pi ) pi ln(1/qi ) = pi ln(1/pi ) − i=1

i=1

i=1

≤

m 

pi (qi /pi − 1)

=

m 

qi −

i=1

i=1

m 

pi

i=1

= 1 − 1 = 0, and equality holds if and only if qi = pi for all i.

Theorem 3.11 The entropy (uncertainty) of a distribution p on m symbols is at most logb m. The maximum value occurs if and only if all the symbols are equally probable.

Proof In Theorem 3.10 take qi = 1/m (1 ≤ i ≤ m). Then Hb (p) ≤

m 

pi logb m = logb m,

i=1

with equality if and only if pi = qi = 1/m for all i (1 ≤ i ≤ m).

EXERCISES 3.10. Suppose a memoryless source emits three symbols a, b, c with probabilities 0.6, 0.3, 0.1, and another memoryless source emits the same symbols, with probabilities 0.5, 0.3, 0.2. Which source has the greater uncertainty? What probability distribution on the three symbols produces the greatest uncertainty? 3.11. Find the derivative h′ (x) of the function h whose graph is shown in Figure 3.1. Verify that the maximum value is at x = 21 and that the tangent becomes vertical as x approaches 0 and 1. 3.12. Interpret the result of Exercise 3.9 in terms of ‘uncertainty’.

38

3. Economical coding

3.5 Optimal codes – the fundamental theorems We now return to the problem proposed in Section 3.2: how to find a UD b-ary code for a source (S, p) such that the average word-length L is minimized. It turns out that the entropy Hb (p) plays a crucial (and rather surprising) part.

Theorem 3.12 The average word-length of any UD b-ary code for the source (S, p) satisfies L ≥ Hb (p).

Proof Denote by yi the length of the codeword for the symbol si , i = 1, 2, . . . , m. Then, as in Section 3.2, the Kraft-McMillan number for the code is K =

1 1 1 + y2 + · · · + ym . by1 b b

Put qi = 1/(Kbyi ) so that q = [q1 , q2 , . . . , qm ] is a probability distribution. Applying the Comparison Theorem 3.10 to p and q we have Hb (p) =

m  i=1

pi logb (1/pi ) ≤

m 

pi logb (1/qi ).

i=1

Since 1/qi = Kbyi , we have logb (1/qi ) = logb K + yi , and so Hb (p) ≤

m  i=1

pi (logb K + yi ) = logb K +

m 

pi yi = logb K + L.

i=1

Since the code is UD, we have K ≤ 1, so logb K ≤ 0 and the result is proved. The obvious question is: how close to the lower bound Hb (p) can L be? In fact it is easy to see that the bound cannot be attained in many cases, for numerical reasons. The last line of the proof of Theorem 3.12 shows that if Hb (p) = L then logb K = 0, so K = 1. Furthermore, it follows from Theorem 3.10 that equality can hold only if pi = qi = 1/Kbyi for all i. In other words, for equality we require byi = 1/pi for all i. However, in general 1/pi is not an integral power of b. For example, in the binary case we can only achieve equality if all the probabilities belong to the set {1/2, 1/4, 1/8, 1/16, . . . }.

3.5 Optimal codes – the fundamental theorems

39

The preceding argument suggests that we can try to construct a code with average word-length close to L by choosing byi to be as close as possible to 1/pi . That is, yi is the least positive integer such that byi ≥ 1/pi . This is sometimes called the Shannon-Fano (SF) rule. The resulting value of K is K =

1 1 1 + y2 + · · · + ym ≤ p1 + p2 + · · · + pm = 1, y 1 b b b

and so (by Theorem 2.9) a PF code with these parameters does exist. The next theorem shows that the average word-length L of such a code is reasonably close to Hb (p).

Theorem 3.13 There is a PF b-ary code for a source with probability distribution p that satisfies the inequality L < Hb (p) + 1.

Proof Let yi (i = 1, 2, . . . , m) be the integer defined by the SF rule, so that a PF code with these parameters exists. We shall show that L < Hb (p) + 1 for this code. Since yi is the least integer such that byi ≥ 1/pi , we have byi −1 < 1/pi . That is, yi − 1 < logb (1/pi ). Hence,  using the fact that pi = 1, we have L=



pi y i
2.)

3.6 Huffman’s rule

41

Lemma 3.15 An optimal PF code c : S → B∗ for a source (S, p) has the properties (1) if the codeword c(s′ ) is longer than c(s) then ps ≥ ps′ ; (2) among the codewords of maximum length there are two of the form x0 and x1, for some x ∈ B∗ .

Proof (1) Suppose the length of w = c(s) is α and the length of w′ = c(s′ ) is α′ . Let c∗ be the code obtained by defining c∗ (s) = w′ and c∗ (s′ ) = w (clearly this is PF). Then the average word-lengths L(c) and L(c∗ ) satisfy L(c∗ ) − L(c) = (ps α′ + ps′ α) − (ps α + ps′ α′ ) = (ps − ps′ )(α′ − α). Since c is optimal, this must be non-negative. Hence if α′ > α then ps ≥ ps′ . (2) If no two words of maximum length have the form stated, then deleting the last bit from all codewords of maximum length would produce a better code that still has the PF property. Huffman’s rule employs two constructions based on these properties (see Figure 3.3). H1 Given a source (S, p), let s′ and s′′ be two symbols with the smallest probabilities. Construct a new source (S ∗ , p∗ ) by replacing s′ and s′′ by a single symbol s∗ , with probability p∗s∗ = ps′ + ps′′ . All other symbols have unchanged probabilities. H2 If we are given a PF binary code h∗ for (S ∗ , p∗ ), with h∗ (s∗ ) = w, then a PF binary code h for (S, p) is defined by the rules h(s′ ) = w0, h(s′′ ) = w1, and h(u) = h∗ (u) for all u = s′ , s′′ .

s′

s′′

H1 :

w0

w1

H2 : s∗

w

Figure 3.3 The two parts of Huffman’s rule

42

3. Economical coding

Suppose we are given a source with m symbols. The rule H1 can be used to construct a sequence of sources each with one symbol less than the previous one, so the process stops at the mth source, which has one symbol. The optimal code for the last source is the trivial one which assigns the empty word to the single symbol. Working backwards, H2 can be used to construct codes for each of the sources in the sequence. The optimality of the resulting codes will be proved in the next section.

Example 3.16 Use Huffman’s rule to construct an optimal code for a source with distribution p = [0.4, 0.2, 0.2, 0.1, 0.1]. Solution Starting with p(1) = p and defining p(i+1) = p(i)∗ for i = 1, 2, 3, 4, the rule H1 produces the sequence of sources shown in Figure 3.4. p(1) :

0.4

0.2

0.2

p(2) :

0.4

0.2

0.2

p(3) :

0.4

0.2

p(4) : p(5) :

0.1

0.1 0.2

0.4

0.6

0.4 1.0

Figure 3.4 Application of rule H1 In order to construct the Huffman code we use H2, starting from the code that assigns the empty word to the single symbol in the last line. The process is shown in Figure 3.5. The codewords are 00, 01, 10, 110, 111. For this source the average word-length of the Huffman code is Lopt = (0.4 + 0.2 + 0.2) × 2 + (0.1 + 0.1) × 3 = 2.2. The entropy H = H(p) is 2.12 approximately. In Example 3.14 the SF rule was applied to this source, and a code with average word-length LSF = 2.8

3.6 Huffman’s rule

43

Figure 3.5 Application of rule H2 was obtained. The relationship between these results and the bounds obtained in Section 3.5 can be summarized as: H < Lopt < LSF < H + 1.

EXERCISES 3.15. Construct an optimal code for the source in Exercise 3.13 using Huffman’s rule, and find its average word-length. 3.16. Consider a source with probability distribution [0.4, 0.3, 0.2, 0.1]. Compare the average word-length of the binary code obtained from the SF rule with the average word-length of the optimal one, obtained by Huffman’s rule. 3.17. Suppose that seven symbols are emitted by a source, with probability distribution p = [0.2, 0.2, 0.2, 0.1, 0.1, 0.1, 0.1]. What is the entropy of the source? Use the Shannon-Fano rule to construct a prefix-free code for this source. Find its average word-length L, and verify that Theorems 3.10 and 3.12 hold. Use Huffman’s rule to construct an optimal binary code for this source.

44

3. Economical coding

3.18. Find the Huffman code for a source with probability distribution [0.4, 0.3, 0.1, 0.1, 0.06, 0.04].

3.7 Optimality of Huffman codes Lemma 3.17 Let h and h∗ be defined as in the construction H2. Then the average wordlengths of h and h∗ satisfy L(h) = L(h∗ ) + p∗s∗ .

Proof Suppose that the length of h∗ (s∗ ) is β. Then in the code h the symbols s′ and s′′ are assigned codewords of length β + 1. Hence L(h) − L(h∗ ) = (ps′ + ps′′ )(β + 1) − p∗s∗ β. Since ps′ + ps′′ = p∗s∗ , this expression reduces to p∗s∗ .

Theorem 3.18 If h∗ is optimal for (S ∗ , p∗ ) then h is optimal for (S, p).

Proof We prove the contrapositive: that is, if h is not optimal then h∗ is not optimal. Suppose h is not optimal, and let c be an optimal code for (S, p). It follows from part (ii) of Lemma 3.15 that there are symbols t′ , t′′ to which c assigns codewords x0 and x1 of maximal length. Suppose first that the pair (t′ , t′′ ) is disjoint from the pair (s′ , s′′ ) involved in the construction of h. Then c(t′ ) = x0, c(t′′ ) = x1, c(s′ ) = y, c(s′′ ) = z, where the codewords have length μ, μ, γ, δ, say. Define a code c∗ for (S ∗ , p∗ ) by c∗ (t′ ) = y, c∗ (t′′ ) = z, c∗ (s∗ ) = x. Here the lengths of the codewords are γ, δ, μ − 1. Hence L(c) − L(c∗ ) = (pt′ + pt′′ )μ + ps′ γ + ps′′ δ − pt′ γ − pt′′ δ − p∗s∗ (μ − 1).

3.7 Optimality of Huffman codes

45

Since p∗s∗ = ps′ + ps′′ , this is can be rewritten as p∗s∗ + (pt′ − ps′ )(μ − γ) + (pt′′ − ps′′ )(μ − δ). Now x0 and x1 are codewords of maximum length, so μ ≥ γ and μ ≥ δ. Also, since s′ , s′′ are symbols with smallest probability, pt′ ≥ ps′ and pt′′ ≥ ps′′ . Hence L(c) − L(c∗ ) ≥ p∗s∗ . We know that L(h) = L(h∗ ) + p∗s∗ (Lemma 3.17), and since h is assumed to be not optimal, L(c) < L(h). Thus L(c∗ ) ≤ L(c) − p∗s∗ < L(h) − p∗s∗ = L(h∗ ), and h∗ is not optimal for (S ∗ , p∗ ). Note that if (t′ , t′′ ) and (s′ , s′′ ) overlap, minor changes are needed.

EXERCISES 3.19. Show that the sum of the ‘new’ probabilities p∗s∗ obtained in the application of rule H1 is equal to L, the average length of the Huffman code. (This result shows that L can be found using only the first part of Huffman’s rule, without having to use H2 to find the code explicitly.) 3.20. Suppose that p is a source with N symbols and and ℓ(C) is the total length of the codewords in an optimal binary code C for p. Show that if C, C (1) , . . . C (N −3) , C (N −2) is the sequence of codes constructed by the Huffman rules, then ℓ(C (N −i−1) ) ≤ ℓ(C (N −i) ) + (i + 1). Deduce that ℓ(C) ≤

1 2 (N + N − 2). 2

3.21. Find the optimal encoding of the source defined in Exercise 3.16 by a ternary code - that is, using the symbols {0, 1, 2}. Justify the assertion that the code is optimal. [It is not necessary to use Huffman’s rule for this exercise.] 3.22. Let m be a positive integer and let pm denote the probability distribution on a set of m symbols in which each symbol is equiprobable: pm = [1/m, 1/m, . . . , 1/m].

46

3. Economical coding

For each m in the range 1 ≤ m ≤ 8, find the optimal binary code for the distribution pm , and its average word-length. On the basis of your results, make a conjecture about the solution for a general m and illustrate it in the case m = 400. Describe the relationship between the entropy H and the average word-length L.

Further reading for Chapter 3 In order to define the concept of a source in full generality it is necessary to use some quite sophisticated probability theory. A good account is given by Goldie and Pinch [3.2]. The Bayesian approach to the subject is comprehensively covered by MacKay [3.4]. The concept of entropy was first studied in theoretical physics, specifically thermodynamics. The connections with information theory were fully explored in the paper by Shannon [3.5], which effectively laid the foundations for the whole subject. The standard text by Cover and Thomas [3.1] is useful. The proof that the entropy function is essentially the only function satisfying a number of conditions that we associate with the idea of uncertainty is given in Appendix 1 of Welsh’s book [3.6]. Huffman’s algorithm [3.3] was discovered when the subject was still in its infancy. 3.1 T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley (Second edition, 2006). 3.2 C.M. Goldie and R.G.E. Pinch. Communication Theory. Cambridge University Press (1991). 3.3 D.A. Huffman. A method for the construction of minimum redundancy codes. Proc. IRE 40 (1952) 1098-1011. 3.4 D.J.C. MacKay. Information Theory, Inference, and Learning Algorithms. Cambridge University Press (2003). 3.5 C.E. Shannon. A mathematical theory of communication. Bell System Tech. J. 27 (1948) 379-423, 623-656. 3.6 D.J.A. Welsh. Codes and Cryptography. Oxford University Press (1988).

4 Data compression

4.1 Coding in blocks We have formalized the idea of a code as a function c : S → T ∗ that replaces symbols in an alphabet S by strings of symbols in an alphabet T . Although we have chosen to regard the elements of S as single objects, such as the letters of a natural alphabet, there is no logical need for this restriction. In practice, it is often useful to split the stream of symbols emitted by a source into blocks (disjoint strings of symbols), and to regard the blocks themselves as symbols. For example, suppose a source emits a stream of letters, each of which is x, y, or z, so that a typical stream would be yzxxzyxzyxzzyxxzyzxzzyyy · · · . Here the alphabet S is the set {x, y, z}. On the other hand, we could split the stream into blocks of length 2: yz xx zy xz yx zz yx xz yz xz zy yy · · · . Now the alphabet is the set S 2 of ordered pairs S 2 = {xx, xy, xz, yx, yy, yz, zx, zy, zz}. In Chapter 3 we considered the problem of coding a source (S, p) so that the average word-length L is as small as possible. We found that the entropy Hb (p) is a lower bound for L, but this bound is rarely attained. We shall now explain how coding in blocks enables us to approach the lower bound as closely as we please. N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 4, 

48

4. Data compression

Example 4.1 Consider a scanner processing a black-and-white document. Suppose it emits W (white pixel) and B (black pixel) with probabilities pW = 0.9, pB = 0.1. Calculate the entropy of this distribution. Assuming that the source is memoryless, find the best binary codes for it using blocks of size 1, 2, and 3. Solution

The entropy is H(p) = 0.9 log2 (1/0.9) + 0.1 log2 (1/0.1) ≈ 0.469.

Using blocks of size 1, the best way to code the source is the obvious one, W → 0, B → 1, because any other code must use a codeword of length greater than 1. This code has average word-length L1 = 1, which is much worse than the lower bound 0.469. A document containing N pixels is encoded by a string of N bits, although theoretically speaking, the amount of information is equivalent to only 0.469N bits. Consider next what happens when we split the message into blocks of size 2. Now we have a source with four ‘symbols’, and the probabilities are: WW 0.81

WB 0.09

BW 0.09

BB . 0.01

What is the optimal binary code for this source? A simple application of Huffman’s rule gives the code W W → 0,

W B → 10,

BW → 110,

BB → 111.

The average length of this code is L2 = 0.81 × 1 + 0.09 × 2 + (0.09 + 0.01) × 3 = 1.29. Observe that a ‘symbol’ now represents two pixels. A document with N pixels will contain N/2 ‘symbols’, and will be encoded by a string with L2 N/2 = 0.645N bits, approximately. Thus we have a significant improvement on our first attempt to achieve the theoretical lower bound of 0.469N . What happens if we use blocks of length 3? Now the source has eight ‘symbols’, and the probabilities are: WWW 0.729

WWB 0.081

W BW 0.081

W BB 0.009

BW W 0.081

BW B 0.009

BBW 0.009

BBB . 0.001

Another application of Huffman’s rule produces the code W W W → 0, W W B → 100, W BW → 101, W BB → 11100, BW W → 110, BW B → 11100, BBW → 11110, BBB → 11111,

4.2 Distributions on product sets

49

for which the average length is L3 = 1.598. Repeating the previous argument, we see that now a document containing N pixels can be encoded by a string of L3 N/3 = 0.533N bits approximately, which is closer still to the theoretical lower bound of 0.469N . The technique described in the example is the basis of data compression. In the rest of this chapter we shall explore its theoretical foundations, and describe some of the coding rules that can be used to implement it.

EXERCISES 4.1. How many words of length ℓ can be formed from an alphabet with r symbols? A message using an alphabet with r symbols has length k. If it contains all possible words of length ℓ as sub-strings (not necessarily disjoint), what is the smallest possible value of k? 4.2. Construct a message that attains the minimum value found in the previous exercise, in the case r = 3, ℓ = 2. 4.3. Consider a memoryless source that emits symbols A and B with probabilities pA = 0.8, pB = 0.2. Calculate the entropy for this source. Find binary Huffman codes for the associated sources using blocks of size 2 and 3, and calculate their average word-lengths L2 and L3 . Can you estimate the limit of Ln /n as n → ∞?

4.2 Distributions on product sets Let S ′ = {s′1 , s′2 , . . . , s′m }, S ′′ = {s′′1 , s′′2 , . . . , s′′n }, and let Y = S ′ × S ′′ be the product set, containing the pairs of symbols s′i s′′j . Let p be a probability distribution on Y , and denote the probability of s′i s′′j by pij .

Definition 4.2 (Marginal distributions, independence) With the notation as above, let p′i =

n  j=1

pij

(i = 1, 2, . . . , m),

p′′j =

m 

pij

(j = 1, 2, . . . , n).

i=1

The distributions p′ on S ′ and p′′ on S ′′ are known as the marginal distributions associated with p. Clearly, p′i is the probability that the first component is s′i ,

50

4. Data compression

and p′′j is the probability that the second component is s′′j . The distributions p′ and p′′ are independent if pij = p′i p′′j .

Example 4.3 Suppose that S ′ = {a, b}, S ′′ = {c, d} and the distribution p is given by the following table. (For example, the entry in row a and column d is pad .) c d 0.3 0.1 0.4 0.2

a b

Find the marginal distributions p′ and p′′ . Are these distributions independent? Solution

We have p′a = pac + pad = 0.4,

p′b = pbc + pbd = 0.6,

p′′c = pac + pbc = 0.7,

p′′d = pad + pbd = 0.3.

The distributions are not independent because (for example) pac = 0.3

whereas

p′a p′′c = 0.28.

Theorem 4.4 The entropies of the distributions p, p′ , p′′ satisfy H(p) ≤ H(p′ ) + H(p′′ ). Equality holds if and only if p′ and p′′ are independent.

Proof By definition H(p′ ) + H(p′′ ) =



p′i log(1/p′i ) +

Since p′i =

j

pij and p′′j =

H(p′ ) + H(p′′ ) = =



i

pij , we have

  i



p′′j log(1/p′′j ).

j

i





i,j

j

pij log(1/p′i ) +

pij log(1/p′i p′′j ).

  j

i

pij log(1/p′′j )

4.2 Distributions on product sets

51

Consider two probability distributions defined on S ′ ×S ′′ : the given distribution p, and q defined by qij = p′i p′′j . Applying the Comparison Theorem (3.10) it follows that   pij log(1/p′i p′′j ), pij log(1/qij ) = H(p) ≤ i,j

i,j

which is just H(p′ ) + H(p′′ ), as shown above. The inequality is an equality if and only if p = q, that is, pij = p′i p′′j , which is the condition that p′ and p′′ are independent. Similar results hold for a product of more than two sets, and they can be proved quite simply by induction. We shall be mainly concerned with the case when all the sets are the same, that is, with the product S r of r ≥ 2 copies of a given set S. In that case an element of S r is just a word of length r in the alphabet S.

Example 4.5 Suppose a source emits a stream of bits, and a number of observations suggest that the frequencies of blocks of length 2 are given by the following probability distribution p on B2 . For example, p01 = 0.4 means that about 40 out of every 100 blocks are 01. 0 1 0 0.1 0.4 1 0.4 0.1 Find the marginal distributions p′ and p′′ , calculate their entropies, and check that Theorem 4.4 holds. Solution The marginal distributions are p′ = [0.5, 0.5] and p′′ = [0.5, 0.5]. Thus H(p′ ) = H(p′′ ) = h(0.5) = 1. On the other hand, H(p) = 0.1 log(1/0.1) + 0.4 log(1/0.4) + 0.4 log(1/0.4) + 0.1 log(1/0.1) ≈ 1.722. Thus Theorem 4.4 holds. In the example the stream has the property that each individual bit is equally likely to be 0 or 1, although the pairs of bits are not equally distributed. The marginal distributions p′ and p′′ are the same, but they are not independent since, for example, p00 = 0.1 whereas p′0 p′′0 = 0.25. In other words the source is not memoryless (Definition 3.2).

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4. Data compression

EXERCISES 4.4. Suppose that X ′ = {u, v, w}, X ′′ = {y, z}, and the distribution p on X ′ × X ′′ is given by the table u v w

y z 0.2 0.1 0.3 0.1 0.1 0.2

Are the marginal distributions p′ and p′′ independent? 4.5. Verify that the entropies of the distributions defined in the previous exercise satisfy Theorem 4.4. 4.6. Suppose it is observed that, in a certain stream of bits, the frequencies of blocks of length 2 are given by the following probability distribution p on B2 . 0 1

0 1 0.35 0.15 0.15 0.35

Find the marginal distributions p′ and p′′ , calculate their entropies, and check that Theorem 4.4 holds.

4.3 Stationary sources In Chapter 3 we considered the stream ξ1 ξ2 ξ3 · · · emitted by a source as a sequence of identically distributed random variables ξk (k = 1, 2, 3 . . .), taking values in a set S. Suppose we consider the stream emitted by a source as a stream of blocks, where each block is a random variable that takes values in the set S r of strings of length r. For example, taking r = 2 we have a stream of random variables ξ2k−1 ξ2k . We should like to define a probability distribution p2 on S 2 by the rule p2 (si sj ) = Pr(ξ2k−1 ξ2k = si sj ) = Pr(ξ2k−1 = si , ξ2k = sj ). Clearly, in order to do this it must be assumed that the probability of emitting the given pair of symbols si sj does not depend on k, the position in the stream. The general form of this condition is as follows.

4.3 Stationary sources

53

Definition 4.6 (Stationary source) A source emitting a stream ξ1 ξ2 ξ3 · · · is stationary if, for any positive integers ℓ1 , ℓ2 , . . . , ℓr , the probability Pr(ξk+ℓ1 = x1 , ξk+ℓ2 = x2 , . . . , ξk+ℓr = xr ) depends only on the string x1 x2 . . . xr , not on k. Although there are many situations where it is reasonable to assume that the condition holds in its general form, in practice we can only check its validity in a few cases. Usually we consider consecutive symbols, that is, the case ℓ1 = 1, ℓ2 = 2, . . . , ℓr = r of the definition. Then the definition implies that for a stationary source we have probability distributions pr , defined on S r for r ≥ 1 by the rule pr (x1 x2 . . . xr ) = Pr(ξk+1 = x1 , ξk+2 = x2 , . . . , ξk+r = xr ) for all k. When r = 1 the definition reduces to our standard assumption (Section 3.1) that all the random variables ξk have the same distribution p1 . Roughly speaking, stationarity means that the probability that a given word (string of consecutive symbols) will appear on ‘page 1’ of the message is the same as the probability that it will appear on ‘page 99’, or any other ‘page’. In practice, we can often check this property experimentally for strings of a certain length r. Then we can use pr to determine the marginal probability distributions ps for 1 ≤ s < r, using the addition law of probability. For instance the distribution on strings of length r − 1 is related to the distribution on strings of length r by the equations   pr (x1 x2 · · · xr−1 s) = pr (sx1 · · · xr−1 ). pr−1 (x1 x2 · · · xr−1 ) = s∈S

s∈S

We have already encountered these equations in the case r = 2. Note that, for a stationary source, it follows that the two marginal distributions associated with p2 are the same. A memoryless source is a very special case of a stationary source. In that case, each pr is simply determined by p1 : pr (x1 x2 . . . xr ) = p1 (x1 )p1 (x2 ) · · · p1 (xr ).

Example 4.7 Consider a stationary source emitting symbols from the alphabet S = {a, b, c},

54

4. Data compression

with the probability distribution p2 on S 2 defined by the following table. a a b c

b

c

0.39 0.17 0.04 0.15 0.11 0.04 0.06 0.02 0.02

Find the corresponding probability distribution p1 on S. Is this source memoryless? What is the relationship between H(p2 ) and H(p1 )? Solution

According to the addition law, p1 (a) = p2 (aa) + p2 (ab) + p2 (ac) = 0.6.

Similarly p1 (b) = 0.3 and p1 (c) = 0.1. The source is not memoryless, since (for example) p2 (aa) = 0.39 whereas p1 (a)p1 (a) = 0.36. The entropy of p1 is 0.6 log(1/0.6) + 0.3 log(1/0.3) + 0.1 log(1/0.1) ≈ 1.295. A similar calculation gives the entropy of p2 as 2.520 approximately. Thus H(p2 ) < H(p1 ) + H(p1 ), in agreement with Theorem 4.4.

EXERCISES 4.7. Suppose that a stationary source emits symbols from the alphabet S = {a, b, c, d} and the probability distribution p2 is given by the table a b c d a b c d

0.14 0.15 0.05 0.06

0.17 0.10 0.02 0.01

0.04 0.04 0.10 0.02

0.05 . 0.01 0.03 0.01

What is the distribution p1 ? Is this a memoryless source? 4.8. Suppose that a source emits a stream of bits, and observations suggest that the frequencies of the blocks of length 2 are given by the following probability distribution on B2 . 0 1

0 0.2 0.3

1 0.3 0.2

4.4 Coding a stationary source

55

In Section 4.2 we noted that this is not a memoryless source. Show that it is not necessarily stationary, by constructing a probability distribution p3 on B3 that can vary in time, but is nevertheless consistent with the observations.

4.4 Coding a stationary source Suppose we regard the stream emitted by a source as a stream of blocks of length r. If the source is stationary then, for each r ≥ 1, there is an associated probability distribution pr , and its entropy H(pr ) is defined. This represents the uncertainty of the stream, per block of r symbols. In order to apply the fundamental theorems relating entropy and average word-length to a stationary source, we must begin with a definition of the entropy of such a source. For blocks of r symbols, the uncertainty per symbol is H(pr )/r, so the uncertainties, measured in bits per symbol, associated with the distributions pr (r ≥ 1) are H(p1 ),

H(p2 ) H(p3 ) , , 2 3

...,

H(pr ) , r

...

.

This set of real numbers has a lower bound 0, since the entropy of any distribution is non-negative. Hence, by a fundamental property of the real numbers, it has a greatest lower bound. This is a number H with the properties (i) H is a lower bound; (ii) no number greater than H is a lower bound. It is customary to refer to the greatest lower bound of a set X as the infimum of the set, written inf X.

Definition 4.8 (Entropy of a stationary source) The entropy H of a stationary source with probability distributions pr is the infimum of the numbers H(pr )/r

r = 1, 2, 3, . . . .

This definition is consistent with the definition of the entropy of a memoryless source (Section 3.2.).

Theorem 4.9 If a stationary source is memoryless, its entropy H is equal to H(p1 ).

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4. Data compression

Proof If the source is memoryless then, by Theorem 4.4, H(p2 ) = 2H(p1 ). In fact, a simple induction proof shows that H(pr ) = rH(p1 ) for all r, and so all the terms H(pr )/r are equal to H(p1 ). In general, the behaviour of the numbers H(pr )/r is constrained by the following consequence of Theorem 4.4.

Lemma 4.10 Suppose n is a multiple of r. Then H(pr ) H(pn ) ≤ . n r

Proof Suppose first that n = 2r. For any ℓ ≥ 1, the distribution of ξℓ+1 ξℓ+2 . . . ξℓ+r and ξℓ+r+1 ξℓ+r+2 . . . ξℓ+2r are pr , by the stationary property. The distribution of ξℓ+1 . . . ξℓ+r ξℓ+r+1 . . . ξℓ+2r is is p2r . Hence, by Theorem 4.4, 1 H(p2r ) H(pr ) ≤ (H(pr ) + H(pr )) = . 2r 2r r A straightforward generalization of the preceding argument shows that, for any q ≥ 2, H(pqr ) ≤ H(p(q−1)r ) + H(pr ). Hence the result follows by induction on q. In practice, it is not feasible to determine the values H(pr )/r for more than a few small values of r. We take the least of these values as an approximation to the actual lower bound H. In theory it is possible to find r so that the approximation is as close as we please, and this leads to a fundamental result: it is possible to construct codes with average word-length (in bits per symbol) arbitrarily close to H.

Theorem 4.11 (The coding theorem for stationary sources) Suppose we have a stationary source emitting symbols from an alphabet S, with entropy H. Then, given ǫ > 0, there is a positive integer n and a prefix-free

4.4 Coding a stationary source

57

binary code for (S n , pn ) for which the average word-length Ln satisfies Ln < H + ǫ. n

Proof Since H is the greatest lower bound for the numbers H(pr )/r, there is an r such that ǫ H(pr ) < H+ . r 2 Choose an integer q such that q > 2/ǫr and let n = qr, so that 1/n < ǫ/2. By Lemma 4.10, H(pn )/n ≤ H(pr )/r. It follows from Theorem 3.13 there is a prefix-free binary code for pn with average word-length Ln < H(pn ) + 1. Hence Ln H(pn ) 1 H(pr ) ǫ < + < + < H + ǫ. n n n r 2

EXERCISES 4.9. Show that the entropy of the source described in Exercise 4.7 does not exceed 1.26 approximately. 4.10. Suppose that a stationary source emits symbols from an alphabet S according to the probability distributions pr on S r (r ≥ 1). Show that H(pa+b ) ≤ H(pa ) + H(pb ) for all a, b ≥ 1. 4.11. Let n and r be such that n = qr + s, where 0 ≤ s < r. Show that H(pr )  s  H(ps ) H(pn ) ≤ . 1− + n r n n Deduce that

H(pn ) = H, n→∞ n where H is the entropy of the source. lim

[Hint. Given ǫ > 0, choose r such that H(pr )/r < H + 21 ǫ, and let K be the maximum value of H(ps ) for 0 ≤ s < r. Hence find a value n0 such that, for all n ≥ n0 , H ≤ H(pn )/n < H + ǫ.]

58

4. Data compression

4.5 Algorithms for data compression We can now see clearly how the technique described in Section 4.1 enables data to be compressed. Suppose a file of N bits is produced by a stationary source for which the entropy is H. Then, according to Theorem 4.11, by coding the file in blocks of sufficient length, we can reduce the size of the file to (H + ǫ)N bits, where ǫ is as small as we please. Huffman’s rule is not well-suited to this method in practice. It requires the construction of codewords for all the blocks before any coding can be done, and if the required block-length is n, the set of codewords will have size 2n .

Example 4.12 Suppose the source is a scan of a black-and-white line drawing, such as the graph of a function (Figure 4.1). Assuming that about 1% of the pixels are black, give an upper bound for the entropy, and estimate the number of codewords that would be required to compress to within 25% of this value.

Figure 4.1 A drawing that defines a source of black and white pixels Solution It is given that H(p1 ) = h(0.01) ≈ 0.08 and so the entropy H of the source is such that H ≤ 0.08. Hence, if we wish to compress to within 25% of the optimum value, we must choose ǫ in Theorem 4.11 to be no larger than 0.02, approximately. To achieve this we must use blocks of length n ≥ 2/ǫ ≈ 100. Thus 2100 codewords may be required. Another weakness of Huffman’s rule is that it assumes that the characteristics of the source are known in advance. Often we wish to update the probabilities in course of processing, so that they better reflect what we know about the source. In fact, Huffman’s rule can be modified to cope with this

4.6 Using numbers as codewords

59

situation, but the modification is not simple. In the light of these considerations there is clearly a need for alternative methods of coding with the following properties: • the codeword for a given block can be calculated in isolation, without the need to find all the codewords; • the codewords can be updated ‘on-line’ (this is known as adaptive coding). In the following sections we shall describe two such methods. The first one, known as arithmetic coding, is based on the idea of assigning short codewords to symbols that occur with high frequencies. Although it is not strictly optimal, it can be proved to be close-to-optimal, and it can be implemented so that the two requirements are satisfied. The second technique, known as dictionary coding, is based on a more heuristic approach, but nevertheless it has been found to work well in practice.

EXERCISES 4.12. Suppose the source is scanning black-and-white diagrams that contain about 10% black pixels. Give an upper bound for the entropy, and estimate the size of the code that would be required to compress to within 25% of this value. 4.13. Consider a memoryless source emitting symbols from the alphabet S = {A, B} with probabilities pA = x, pB = 1 − x. Let L2 (x) denote the average word-length of the associated Huffman code on S 2 . Draw the graph of L2 (x) for 0 ≤ x ≤ 1, noting any significant features.

4.6 Using numbers as codewords The technique known as arithmetic coding is based on a correspondence between binary words (elements of B∗ ) and fractions (elements of the set Q of rational numbers). For any word z1 z2 · · · zn in Bn there is a rational number z2 z1 zn + 2 + ···+ n. 2 2 2 This number lies between 0 and 1, and it is denoted by 0.z1 z2 . . . zn in binary notation. Thus, for each integer n ≥ 1 we have a function Bn → Q, defined by z1 z2 · · · zn → 0.z1 z2 . . . zn .

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4. Data compression

Example 4.13 Write down the rational numbers corresponding to the words 0110101 and 1010000. Solution Here n = 7. The word 0110101 corresponds to the rational number 0.0110101 =

1 1 53 1 1 + + + = , 4 8 32 128 128

or 0.4140625 in the usual (decimal) notation. Similarly, the word 1010000 corresponds to the rational number 1 1 5 + = , 2 8 8 or 0.625 in decimal notation. Note that the final 0’s are normally omitted, so the binary form would be written as 0.101. But when it is important to record the value of n, the final 0’s in the binary notation must be shown. 0.1010000 =

In arithmetic coding we use rational numbers to define codewords representing strings of symbols. The aim is to obtain codes that are close to optimal. As usual we do this by trying to ensure that a string X with high probability is represented by a codeword c(X) with small length n. The codewords will therefore be defined in terms of the probabilities. We begin with the set R of real numbers, because the probability of an event is usually allowed to be a real number, rather than a rational number. We shall choose the codeword c(X) to be a number in a suitable interval of the form [a, a + P ) = {r ∈ R | a ≤ r < a + P }, where a is a real number and P is the probability of X. The next theorem explains how we can pick a rational number of the form 0.z1 z2 . . . zn , however small the length P of the interval.

Theorem 4.14 Suppose a and P are such that 0 ≤ a < a + P ≤ 1. Let n be any integer such that 2n > 1/P . Then there is a word z1 z2 . . . zn ∈ Bn such that 0.z1 z2 . . . zn ∈ [a, a + P ).

Proof The interval [0, 1) is partitioned into 2n disjoint intervals Ji of length 1/2n , where   i−1 i , i = 1, 2, . . . , 2n . Ji = 2n 2n

4.6 Using numbers as codewords

61

The given number a is in exactly one of these intervals, say Jc (Figure 4.2).

Figure 4.2 a is in the interval Jc Equivalently, c is the unique integer such that c − 1 ≤ 2n a < c. We are given that a + P ≤ 1 and 2n > 1/P , hence c = (c − 1) + 1 < 2n a + 2n P = 2n (a + P ) ≤ 2n . Thus c < 2n and the binary representation of c has the form c = 2n−1 z1 + 2n−2 z2 + · · · + zn . Finally, we have shown that c > 2n a and c < 2n (a + P ), so c/2n = 0.z1 z2 . . . zn lies in the interval [a, a + P ) (Figure 4.3.)

a

a+P

0

1 c 2n

Figure 4.3 c/2n is in the interval [a, a + P )

62

4. Data compression

EXERCISES 4.14. Find the rational numbers corresponding to the words 0111010 and 1001001. 4.15. Explain why the rational number that we normally write as 1/3 does not correspond to a binary word z1 z2 . . . zn for any n ∈ N.

4.7 Arithmetic coding We shall use Theorem 4.14 to construct prefix-free binary codes for the strings of symbols that are emitted by a stationary source. In that case there is a probability distribution on the set S r of strings X = x1 x2 · · · xr of length r in S. (In order to avoid cumbersome notation we shall denote the distribution on S r by P , for any value of r.) In general we do not assume the independence property P (x1 x2 · · · xr ) is not necessarily equal to P (x1 )P (x2 ) · · · P (xr ). We shall suppose that the symbols in the alphabet S are arranged in a fixed order α < β < γ < δ < · · · < ω. Using this order we can define an order on the strings of symbols, in the same way as the words in a dictionary are ordered using the alphabetical order of the letters.

Definition 4.15 (Dictionary order) The dictionary order on S r is defined as follows. If X, Y are two different strings in S r , let i be the least integer such that xi = yi . If yi < xi put Y < X, otherwise put X < Y . It is convenient to reserve α and ω for the first and last symbols, irrespective of the size of S. For example, if |S| = 3, we take then S = {α, β, ω}, with the order α < β < ω. In this case the number of strings of length 4 is 34 = 81. The first seven strings are αααα < αααβ < αααω < ααβα < ααββ < ααβω < ααωα < . . . and the last seven are . . . < ωωαω < ωωβα < ωωββ < ωωβω < ωωωα < ωωωβ < ωωωω.

4.7 Arithmetic coding

63

Definition 4.16 (Cumulative probability function) Given a distribution P on S r we define the associated cumulative probability function a on S r as follows. If X is the first string in S r put a(X) = 0. Otherwise put  P (Y ). a(X) = Y m? In this ‘awkward case’ r can only be m + 1, because c2 is constructed using the

70

4. Data compression

dictionary D1 . According to the rule for Step 1, dm+1 = sp sq where x1 = sp , x2 = sq . Now the decoder can identify x2 , because c1 c2 is the encoded form of sp sp sq . Thus x2 = sp so q = p and the new entry in the dictionary is actually dm+1 = sp sp . A similar argument works at every step, as in the proof of the following theorem.

Theorem 4.24 An LZW code constructed as in Definition 4.22 is uniquely decodable.

Proof Suppose that the partial code c1 c2 . . . ck−1 has been successfully decoded as x1 x2 . . . xi . Suppose also that the dictionary Dk−1 has been constructed by adding k − 1 strings to D0 . The argument given above shows that these assumptions are justified when k − 1 = 1, so we may suppose that k ≥ 2. We must give the rules for decoding ck and constructing Dk = (Dk−1 , dm+k ). The encoding rules imply that ck is the index of a string su . . . sv in Dk−1 . Thus ck is decoded by putting xi+1 = su , . . . , xj = sv . In order to construct the new dictionary entry dm+k , the encoder must use the value of ck+1 . If ck+1 = r ≤ m + k − 1 then dr is in Dk−1 , say dr = sa . . . . Thus xi+1 = sa , and the new dictionary entry is dm+k = su . . . sv sa . Since the construction of ck+1 uses only Dk , the only other possibility is the ‘awkward case’ ck+1 = m + k. Now dm+k is a string of the form su . . . sv sz , where sz = xj+1 . So in this case ck ck+1 is the encoded form of su . . . sv su . . . sv sz , and in fact xj+1 = su . In other words dm+k = su . . . sv su . This completes the emulation of Step k, and the proof is complete.

Example 4.25 Given the dictionary D0 = I,M,P,S, decode the message 214468331

.

Solution At Step 1, c1 = 2 and d2 = M, so x1 = M. Also c2 = 1 and d1 = I, so the new dictionary entry is MI. Thus 2 → M,

D1 = I, M, P, S, MI.

4.9 Coding with a dynamic dictionary

71

At Step 2, c2 = 1 and d1 = M, so x2 = I. Also c3 = 4 and d4 = S, so the new dictionary entry is IS. Thus 2 1 → MI,

D2 = I, M, P, S, MI, IS.

Steps 3 and 4 are similar, resulting in 2 1 4 4 → MISS,

D4 = I, M, P, S, MI, IS, SS, SI.

At the next step c5 = 6, and d6 = IS so x5 = I, x6 = S. Also c6 = 8 and d8 = SI so the new dictionary entry is ISS. Thus 2 1 4 4 6 → MISSIS,

D5 = I, M, P, S, MI, IS, SS, SI, ISS.

Steps 7,8,9 are similar, and the result is 2 1 4 4 6 8 3 3 1 → MISSISSIPPI. At this point, the reader is probably asking whether the LZW rules actually achieve significant compression. In the toy examples given above, only a very small amount of compression is achieved, at some cost. For example, the message ABRACADABRA has length 11, and the coded form 1 2 5 1 3 1 4 6 8 has length 9, but the reduction is achieved by using a nontrivial calculation. However the examples do suggest that, as the message increases in length, with inevitably more repetition of strings, then there may be some improvement. In practice, LZW coding works well, and its widespread use confirms that the principle is sound. The presentation of the rules given above is intended to make them easily understood, but it is possible to streamline the calculations by using more efficient data structures.

EXERCISES 4.23. Verify that the LZW encoding rules for MISSISSIPPI with the initial dictionary I,M,P,S produce the code 2 1 4 4 6 8 3 3 1. 4.24. Verify that the LZW decoding rules for 1 2 5 1 3 1 4 6 8 with the initial dictionary A,B,C,D,R produce the message ABRACADABRA. 4.25. Suppose the initial dictionary is B,D,E,N,O,R,T, ⊔. Decode the following message. 1 3 7 8 5 4 8 7 3 14 5 6 8 9 11 13 12 4 3 12 20 2 25 5 11 22 4.26. Suppose we are given D0 = a, b, c and the coded message 1 2 1 3 4 7 9. Show that at Step k = 6 of the decoding procedure we encounter the ‘awkward case’ in which the codeword ck+1 is not the index of an entry in the dictionary Dk−1 . Explain how the difficulty is resolved.

72

4. Data compression

Further reading for Chapter 4 The basic ideas of arithmetic coding were known to the founding fathers of information theory, but it was not until the advances in computer technology in the 1970s that there was serious interest in data compression techniques. The influential paper by Gallager [4.1] discussed various possible methods of adaptive coding. Slightly earlier, Jacob Ziv and Abraham Lempel [4.4] had suggested the idea of coding with a dictionary. There are several variants of their idea; the one known as LZW that is discussed in this book was developed by Welch in 1984 [4.3]. The LZW algorithm is now used in many applications, including compress, gzip, and Acrobat. A standard reference for the practical side of data compression is the book by Sayood [4.2], and MacKay [3.3, Chapter 6] gives an interesting commentary on the respective merits of the various approaches. In this book we discuss stationary sources because they provide the simplest model that occupies the common ground between theory and practice. There are more sophisticated models, such as a Markov source and an ergodic source, and these too have been the subject of serious study by mathematicians. An excellent introduction is given by Welsh [3.5]. It is traditional to define the entropy H of a stationary source to be the limit of the sequence H(pr )/r, rather than the infimum. This creates a problem, because it is not obvious that the limit exists, whereas the existence of the infimum is a basic property of the real number system. In fact the two quantities are equal, but the proof is rather complicated (see Exercise 4.11). All the important properties of H can be proved using the simpler definition. 4.1 R.G. Gallager. Variations on a theme by Huffman. IEEE Trans. Info. Theory IT-24 (1978) 668-674. 4.2 N. Sayood. Introduction to Data Compression, Third edition. MorganKaufmann (2005). 4.3 T.A. Welch. A technique for high performance data compression. IEEE Computer 17 (1984) 8-19. 4.4 J. Ziv and A. Lempel. A universal algorithm for sequential data compression. IEEE Trans. Info. Theory IT-23 (1977) 337-343.

5 Noisy channels

5.1 The definition of a channel Let I be a finite alphabet. We consider the symbols in I as inputs to a device, referred to as a channel, which transmits them in such a way that errors may occur. As a result of these errors, which we call noise, the symbols that form the output of the device belong to another set J. The sets I and J may be the same, but that does not mean that a specific input i ∈ I will result in the output of the same symbol. This situation is illustrated in Figure 5.1.

Inputs I

Channel

Outputs J

Figure 5.1 A diagrammatic representation of a ‘channel’ For each i ∈ I and j ∈ J, we shall by denote by Pr(j | i) the conditional probability that the output is j, given that the input is i: Pr(j | i) = Pr(output is j | input is i). N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 5, 

74

5. Noisy channels

Definition 5.1 (Noisy channel) A channel Γ with input set I and output set J is a matrix whose entries are the numbers Pr(j | i): Γij = Pr(j | i)

(i ∈ I, j ∈ J).

(Note that the entries of Γ are labelled so that the rows correspond to inputs and the columns correspond to outputs.) If at least one of the terms Γij with i = j is non-zero, we say that the channel is noisy. From a mathematical viewpoint, there is no distinction between a channel and the corresponding channel matrix, and we shall use these terms interchangeably.

Definition 5.2 (Binary Symmetric Channel) A Binary Symmetric Channel (BSC) corresponds to a matrix of the form     Γ00 Γ01 1−e e Γ = = . e 1−e Γ10 Γ11

0

1−e

0

e e 1

1−e

1

Figure 5.2 The binary symmetric channel with bit-error probability e A Binary Symmetric Channel is illustrated in Figure 5.2. The rows and columns of the channel matrix Γ correspond to 0 and 1, the elements of the binary alphabet B. The fact that Γ01 = Γ10 = e means that the output symbol differs from the input symbol with probability e, and we refer to e as the bit-error probability. The probability that a symbol is transmitted correctly is Γ00 = Γ11 = 1 − e. Usually we assume that e is a small positive number: for example, e = 0.01 means that one bit in every hundred is transmitted wrongly. Of course, that would be unacceptably large in most real situations. Here is a different kind of channel.

5.1 The definition of a channel

75

Example 5.3 Figure 5.3 illustrates a simplified keypad with six keys A, B, C, D, E, F . Let us say that two keys are adjacent if an edge of one is next to an edge of the other. Given any two adjacent keys x and y there is a probability 0.1 that when I intend to enter x I shall press y. Write down the channel matrix for this situation. If I intend to enter F ACE, what is the probability that the keypad will register correctly?

A

B

C

D

E

F

Figure 5.3 A simplified keypad Solution matrix is

The input and output sets are {A, B, C, D, E, F }, and the channel 0.8 ⎜ 0.1 ⎜ ⎜ 0 Γ = ⎜ ⎜ 0.1 ⎜ ⎝ 0 0 ⎛

0.1 0.7 0.1 0 0.1 0

⎞ 0 0.1 0 0 0.1 0 0.1 0 ⎟ ⎟ 0.8 0 0 0.1 ⎟ ⎟. 0.0 0.8 0.1 0 ⎟ ⎟ 0 0.1 0.7 0.1 ⎠ 0.1 0 0.1 0.8

The probability of correctly registering F ACE is 0.8 × 0.8 × 0.8 × 0.7 = 0.3584.

Lemma 5.4 Each row of a channel matrix has sum equal to 1. That is,  Γij = 1 for all i ∈ I. j∈J

Proof By definition,  j∈J

Γij =

 j∈J

Pr(j | i).

76

5. Noisy channels

The right-hand side represents the total probability of all outputs, given that the input is i, and so it is equal to 1.

EXERCISES 5.1. A binary asymmetric channel is similar to a BSC, except that the probability of error when 0 is sent is a, and the probability of error when 1 is sent is b, where a = b. Write down the matrix for this channel. 5.2. A ternary symmetric channel is a channel for which the input set I and output set J are both {0, 1, 2}, and the probability that a symbol i becomes j = i is x. Write down the matrix for this channel.

5.2 Transmitting a source through a channel Suppose that a symbol i in the input set I occurs with probability pi , so that we have a probability distribution p on I. Then we can think of the input to the channel Γ as being generated by a source (I, p). Similarly, letting qj be the probability that the output symbol is j, the output can be regarded as a source (J, q). The following theorem describes the relationship between p and q.

Theorem 5.5 Let Γ be a channel matrix, and let the distributions associated with the input source (I, p) and the output source (J, q) be written as row vectors, p = [p1 , p2 , . . . , pm ],

q = [q1 , q2 , . . . , qn ].

Then q = pΓ.

Proof For each i ∈ I and j ∈ J let tij denote the probability of the event that the input to Γ is i and the output is j. According to the addition law, it follows that qj =

 i∈I

tij .

5.2 Transmitting a source through a channel

77

Also, by the definition of conditional probability, tij = Pr(output is j | input is i) × Pr(input is i) = Γij pi . Thus qj =



pi Γij = (pΓ )j ,

i∈I

that is, q = pΓ.

We now have a model in which the channel Γ can be regarded as a link between two sources, (I, p) and (J, q). The first source is produced by a Sender, while the second source, the result of transmission through Γ , is available to a Receiver. These sources are related by the equation q = pΓ .

Example 5.6 Write down explicitly the equations linking p = [p0 , p1 ] and q = [q0 , q1 ] when Γ is the BSC with bit-error probability e. If e = 0.1 and p = [0.7, 0.3], what is q? Solution

The matrix equation is ( q0

q1 ) = ( p0

which is equivalent to the equations q0 q1

p1 )



1−e e e 1−e



,

= p0 (1 − e) + p1 e = p0 e + p1 (1 − e).

If e = 0.1 and p = [0.7, 0.3] then q = [0.66, 0.34].

EXERCISES 5.3. Suppose the output symbols from a channel Γ1 are the input symbols for channel Γ2 , and Γ is the resulting combined channel. In this situation we say that Γ is the result of combining Γ1 and Γ2 in series. What is the relationship between the matrices for Γ1 , Γ2 , and Γ ?

78

5. Noisy channels

5.4. Suppose that a BSC with bit-error probability e and a BSC with bit-error probability e′ are combined in series. Show that the result is a BSC, and find its bit-error probability. 5.5. Suppose that n binary symmetric channels with the same bit-error probability e are combined in series. If 0 < e < 1, what is the ‘limit’ channel as n → ∞? What happens if e = 0? What happens if e = 1? 5.6. Consider the binary symmetric channel with bit-error probability e = 0.01. If the input has probability distribution p = [0.6, 0.4] what is the output distribution q? Compare the entropies of the input and output. 5.7. Consider a binary asymmetric channel (Exercise 5.1) with a = 0.02, b = 0.04. Write down explicitly the equations for q in terms of p. 5.8. It is observed that in the output from a binary asymmetric channel with a = b the symbols 0 and 1 occur equally often. Find the probability distribution on the input, and show that the entropy of the output exceeds that of the input.

5.3 Conditional entropy The existence of errors tends to equalize probabilities, because symbols that occur more frequently are transmitted wrongly more often. In Example 5.6 the uncertainty of the input and the output are, respectively, h(0.7) ≈ 0.881,

h(0.66) ≈ 0.925,

where h is the standard entropy function. As we might expect, uncertainty is increased by transmission through a noisy channel. More precisely, from the viewpoint of an observer who has access to both input and output, the effect of transmitting a source through a noisy channel Γ is to increase the uncertainty. A more subtle problem is to describe the situation from the viewpoint of the Receiver, for whom the output provides the only available information about the input. Question: How much information about the input is available to a Receiver who knows the output of Γ ? In order to answer this question, it is helpful to reconsider the situation. We set up the model in terms of an input that is passed through a channel to produce an output, because that is the way an engineer would think of it. But

5.3 Conditional entropy

79

from a mathematical viewpoint the fundamental object is simply a probability distribution t on the set I × J: tij = Pr(input is i and output is j). We have already used t in the proof of Theorem 5.5. The significant point is that, when t is given, all the other quantities can be derived from it, by using the equations   tij , Γij = tij /pi . tij , qj = pi = i

j

In particular, p and q are the marginal distributions associated with t.

Definition 5.7 (Conditional entropy) With the notations as above, the conditional entropy H(p | q) is defined to be H(p | q) = H(t) − H(q). The motivation is that H(t) measures the uncertainty about the inputoutput pair (i, j), and H(q) measures the uncertainty about the output j. Thus, subtracting the second quantity from the first represents the uncertainty of a Receiver who knows the output and is trying to determine the input. It is worth noting that an alternative definition, more complicated but equivalent, is often used for H(p | q) – see Section 5.5, particularly Exercise 5.18. Since the input and output distributions are linked by the equation q = pΓ , it follows that H(p | q) depends only on Γ and p. We shall often refer to this quantity as the conditional entropy of p with respect to transmission through Γ , and denote it by H(Γ ; p): H(Γ ; p) = H(p | q)

(where q = pΓ ).

Example 5.8 Let Γ be the BSC with bit-error probability e = 0.1, and let the source distribution be p = [0.7, 0.3]. Calculate H(Γ ; p). Solution

Since tij = pi Γij we have t00 = 0.63,

t01 = 0.07,

t10 = 0.03, t11 = 0.27.  It follows that H(t) ≈ 1.350. Since qj = i tij we have q = [0.66, 0.34], and H(q) ≈ 0.925. Hence H(Γ ; p) = H(t) − H(q) ≈ 1.350 − 0.925 = 0.425.

80

5. Noisy channels

The general result for a binary symmetric channel is as follows.

Theorem 5.9 Let Γ be the BSC with bit-error probability e, and let p be the source distribution [p0 , p1 ] = [p, 1 − p]. Then H(Γ ; p) = h(p) + h(e) − h(q), where q = p(1 − e) + (1 − p)e, and h is the standard entropy function defined by h(x) = x log2 (1/x) + (1 − x) log2 (1/(1 − x)).

Proof The values of tij = pi Γij are t00 = p(1 − e),

t01 = pe,

t10 = (1 − p)e,

t11 = (1 − p)(1 − e).

These values mean that the probability distribution t on B2 is simply the product of independent distributions [p, 1 − p] and [1− e, e]. Hence, by Theorem 4.4, H(t) is the sum of the entropies of these distributions, and H(Γ ; p) = H(t) − H(q) = h(p) + h(e) − h(q).

EXERCISES 5.9. Consider the binary symmetric channel with bit-error probability e = 0.01 and input distribution p = [0.6, 0.4] (Exercise 5.6). Calculate the joint distribution t and hence find the conditional entropy H(Γ ; p). 5.10. Verify that your answer to the previous exercise agrees with the formula in Theorem 5.9. 5.11. The binary erasure channel accepts input symbols 0, 1. The output symbol is the same as the input symbol with probability c and a query (?) with probability 1 − c. In other words, the channel matrix is   c 0 1−c Γ = . 0 c 1−c If the input distribution is p = [p, 1 − p], show that

5.4 The capacity of a channel

81

H(Γ ; p) = (1 − c)h(p). 5.12. Consider a general binary channel with I = J = B. If the joint distribution on I × J is given by t00 = a,

t01 = b,

t10 = c,

t11 = d,

a + b + c + d = 1,

write down p, q, and Γ , and verify that q = pΓ .

5.4 The capacity of a channel It is fairly obvious that an oil pipeline or a road has a ‘capacity’, beyond which it cannot function effectively. But in the case of a communication channel this feature is not so obvious. In fact, a good definition does exist, and significant results can be proved about it. We have defined • H(p) : the uncertainty about symbols emitted by the input source p; • H(Γ ; p) : the uncertainty about symbols emitted by p from the viewpoint of a Receiver who knows the symbols emitted by the output source q = pΓ . The following theorem confirms that these quantities are related in the way that we should expect.

Theorem 5.10 Let Γ be a channel and p an input distribution for Γ . Then H(Γ ; p) ≤ H(p). Equality holds if and only if p and q = pΓ are independent distributions.

Proof The distributions p and q are the marginal distributions associated with the joint distribution t. It follows from Theorem 4.4 that H(t) ≤ H(p) + H(q), with equality if and only if p and q are independent. By definition, H(Γ ; p) = H(t) − H(q), and so H(Γ ; p) + H(q) = H(t) ≤ H(p) + H(q), with equality if and only if p and q are independent. Hence the result.

82

5. Noisy channels

Let fΓ (p) = H(p) − H(Γ ; p). Since fΓ (p) is a difference between two measures of uncertainty, we can think of it as a measure of information. Specifically, it represents the information about symbols emitted by the input source p that is available to the Receiver who knows the symbols emitted by the output source. For example, suppose Γ is the BSC with bit-error probability e = 0.1, and the input distribution is p = [0.7, 0.3]. In Section 5.3 we found that H(p) = h(0.7) ≈ 0.881 and H(Γ ; p) ≈ 0.425, so fΓ (p) = H(p) − H(Γ ; p) ≈ 0.881 − 0.425 = 0.456. Suppose we are given a channel Γ that accepts an input alphabet I of size m; in other words, the channel matrix has m rows. Then for each probability distribution p on I we have a value of fΓ (p).

Definition 5.11 (Capacity) The capacity γ of Γ is the maximum value of fΓ (p), taken over the set P of all probability distributions on a set of size m (the number of rows of Γ ). That is,   γ = max fΓ (p) = max H(p) − H(Γ ; p) . P

P

It is important to stress that the maximum exists. Technically this is because fΓ is a continuous function, and the set of all probability distributions, P = {p ∈ Rm | p1 + p2 + · · · + pm = 1, 0 ≤ pi ≤ 1}, is a compact set. For example, when m = 3 the set P is the subset of R3 indicated by the shaded area in Figure 5.4. p3

p2

p1

Figure 5.4 The set P when m = 3

5.5 Calculating the capacity of a channel

83

The definition of capacity is fundamental. Its full significance will appear in the following chapters, when we discuss some famous results about the transmission of coded messages through noisy channels.

EXERCISES 5.13. Find the maximum values of the following functions on the set P when m = 3. (i) p1 p2 p3 ;

(ii) p21 + p22 + p23 ;

(iii) p1 + 2p2 + 3p3 .

5.14. If p = [p1 , p2 , . . . , pm ], find the maximum value of H(p) on the set P. [Calculus methods are not required.]

5.5 Calculating the capacity of a channel In general, calculating the capacity of a channel is difficult, but for the BSC there is a simple answer.

Theorem 5.12 The capacity of the BSC with bit-error probability e (0 ≤ e < 21 ) is γ = 1 − h(e), where h(e) = e log2 (1/e) + (1 − e) log2 (1/(1 − e)).

Proof Writing p = [p, 1 − p] we have H(p) = h(p), and according to Theorem 5.9 H(Γ ; p) = h(p) + h(e) − h(q), where q = p(1 − e) + (1 − p)e. Thus the capacity is the maximum of   h(p) − h(p) + h(e) − h(q) = h(q) − h(e) taken over the range 0 ≤ p ≤ 1.

84

5. Noisy channels

Since e is a given constant, the maximum occurs when the first term h(q) is a maximum. This occurs when q = 21 , that is q = p(1 − e) + (1 − p)e =

1 . 2

Rearranging this equation we get 1 (p − )(1 − 2e) = 0. 2 So, for any e < 12 , the maximum of h(q) occurs when p = 21 , and the maximum value is h( 12 ) = 1. The capacity is therefore   γ = max h(q) − h(e) = 1 − h(e).

In most real situations, e is small, so h(e) is close to 0, and γ is close to 1. For example, the capacity of the BSC with bit-error probability 0.01 is γ = 1 − h(0.01) = 1 − 0.01 log2 (1/0.01) − 0.99 log2 (1/0.99) = 0.919. The definition of channel capacity (Definition 5.11) is formulated in terms of finding the maximum of H(p) − H(Γ ; p),

that is

H(p) − H(p | q).

We chose this formulation because it motivates the interpretation of capacity as the maximum amount of information that can be transmitted. However, there are other formulations that are mathematically equivalent, and sometimes more convenient for the purposes of calculation. Since H(p | q) = H(t) − H(q), the quantity to be maximized can be expressed more symmetrically as H(p) + H(q) − H(t). By analogy with the definition of H(p | q) as H(t) − H(q) we can define H(q | p) to be H(t) − H(p). Then the quantity to be maximized is H(q) − H(q | p). We shall describe an alternative way of calculating H(q | p). According to Lemma 5.4, for each input symbol i the numbers Γij (j ∈ J) define a probability distribution on the set of output symbols. The entropy of this distribution is

5.5 Calculating the capacity of a channel

85

a measure of the uncertainty about the output symbol, given that the input symbol is i, and we shall denote it by H(q | i) =



Γij log(1/Γij ).

j∈J

The next theorem proves that H(q | p) is the average of these uncertainties, taken over the the set of input symbols.

Theorem 5.13 Let H(q | p) = H(t) − H(p). Then H(q | p) =

 i∈I

pi H(q | i).

Proof Let S stand for the sum on the right-hand side. Then we have   pi log(1/pi ) pi H(q | i) + S + H(p) = i

i

=

 i

  pi H(q | i) + log(1/pi ) .

 Using the definition of H(q | i) and the fact that j Γij = 1, it follows that   Γij log(1/pi ) Γij log(1/Γij ) + H(q | i) + log(1/pi ) = j

j

=



Γij log(1/pi Γij ).

j

Since pi Γij = tij , this simplifies to 1  tij log(1/tij ). pi j Hence S + H(p) =



tij log(1/tij ) = H(t),

i,j

and so S = H(t) − H(p) = H(q | p), as claimed.

86

5. Noisy channels

Using this result, we can calculate the capacity γ using the alternative formula   where q = pΓ. γ = max H(q) − H(q | p) P

Example 5.14 Find the capacity of a channel represented by a matrix of the form   s t t s , t s s t where 2(s + t) = 1. Solution

The capacity is the maximum of H(q) − H(q | p), where H(q | p) = p1 H(q | 1) + p2 H(q | 2).

The entropies on the right-hand side can be calculated directly from the matrix Γ: H(q | 1) = H(q | 2) = 2(s log(1/s) + t log(1/t)). Since p1 + p2 = 1, H(q | p) has the same value. This value is independent of q, so the capacity is obtained when H(q) is a maximum, that is, when q = ( 41 , 14 , 14 , 14 ), and H(q) = log 4. Hence the capacity is log 4 − 2(s log(1/s) + t log(1/t)) = 2(1 + s log s + t log t).

EXERCISES 5.15. A channel Γ accepts input symbols 1, 2, . . . , 2N and produces the output 0 when the input is an even number, and 1 when the input is an odd number. Write down the matrix that represents this channel. Show that for any input distribution p H(p) − H(Γ ; p) = H(q), where q = pΓ . Hence show that the capacity of this channel is 1 and explain this result. 5.16. Consider a binary asymmetric channel with matrix   1−a a Γ = . b 1−b

5.5 Calculating the capacity of a channel

87

Let the input and output sources be p and q = [q, 1 − q]. Show that fΓ (p) can be written as a function of q in the form F (q) = h(q) − (qx1 + (1 − q)x2 ), where x1 and x2 satisfy the equation     h(a) x1 = . Γ h(b) x2 Noting that x1 and x2 do not depend on the source, deduce that the capacity of the channel is log2 (2x1 + 2x2 ). 5.17. A simplified keypad has four keys arranged in two rows of two (compare Figure 5.3). If the intention is to press key x, there is probability α of pressing the other key in the same row and probability α of pressing the other key in the same column (and consequently probability 1 − 2α of pressing x). Write down the channel matrix for this situation and find its capacity. 5.18. In Section 5.3 we gave the formulae for deriving pi , qj , and Γij from the joint distribution t. Verify that, for each j ∈ J, the formula Δij = tij /qj = Pr(input is i | output is j) defines a probability distribution on I. The entropy of this distribution is denoted by H(p | j). Using the same method as in the proof of Theorem 5.13, show that  H(p | q) = qj H(p | j). j

[This is often used as the definition of H(p | q). The numbers Δij are known as inverse probabilities.]

Further reading for Chapter 5 The concepts of a channel and its capacity were formulated by Shannon in his fundamental paper, referenced at the end of Chapter 3 [3.4]. This paper was later republished as a book, with a useful introduction by Weaver [5.3]. There are several books that develop the theory at a more abstract level, including those by Ash [5.1] and McEliece [5.2]. 5.1 R. Ash. Information Theory. Wiley, New York (1965).

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5.2 R.J. McEliece. The Theory of Information and Coding. Addison-Wesley, Reading, Mass. (1977). 5.3 C.E. Shannon and W. Weaver. The Mathematical Theory of Communication. University of Illinois Press, Urbana (1963).

6 The problem of reliable communication

6.1 Communication using a noisy channel In this chapter we consider the transmission of coded messages through a noisy channel. This is the setting for some very significant results. The general situation is quite complex, and for clarity of exposition we shall consider messages coded in the binary alphabet B = {0, 1}, with the symbols being transmitted through a binary symmetric channel. However, it is worth noting that the main results (to be discussed in the next chapter) also hold more generally. We consider a source emitting a stream of symbols belonging to an alphabet X. The symbols will be encoded as binary words, all having the same length n, where n is going to be chosen so that some desirable conditions are satisfied. For example, suppose the source is a controller guiding a robot through a network of streets, running North-South or East-West. Some streets are blocked by parks and lakes, and some are one-way. In order to move the robot from one location to another, the controller must send a sequence of the symbols N, S, E, W. Each time the robot arrives at a junction it will use the relevant symbol to decide which direction to take (see Figure 6.1). Here the alphabet is X = {N, S, E, W}. Taking n = 2, a suitable code c : X → B2 would be N → 00,

S → 01,

E → 10,

W → 11.

Using this code the sequence of instructions shown in Figure 6.1 is encoded as follows: N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 6, 

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6. The problem of reliable communication

LAKE

Figure 6.1 The route E,N,N,E,S,E,E,S,W

(E, N, N, E, S, E, E, S, W) → 100000100110100111 . The value n = 2 is clearly the smallest possible value for a binary code with four messages. However, this code has the disadvantage that if any error whatsoever is made when the bits are transmitted, the robot will not be able to detect it. For example, suppose the intended symbol is S, so that 01 is sent, and the first bit is altered in transmission, so that 11 is received. The robot will decode this as W, and has no means of knowing that a mistake has been made. The example illustrates the following scenario. Messages originate from some real source and are expressed in an alphabet X. We shall refer to the output from this source as the original stream. A Sender must transmit these messages to a Receiver, using a binary symmetric channel. In order to do this the Sender encodes the original stream using a binary code c : X → Bn . We shall refer to the stream emitted by the Sender as the encoded stream. The encoded stream is a string of codewords, each of which is a binary word of length n, and so it is in fact a string of bits. In summary, the Sender has effected a transformation T1:

original stream −→ encoded stream.

Note that the encoded stream is based on a code which is prefix-free, since all codewords have the same length n. Thus, at this stage, the decoding problem is simple. However, when the bits are transmitted through a BSC, errors are made. The output of the channel, which we shall refer to as the received stream, is not the same as the encoded stream. In other words the process of transmission has effected a transformation

6.1 Communication using a noisy channel

91

T2: encoded stream −→ received stream. We may assume that the Receiver has a ‘codebook’, so that the set of codewords C and the word-length n are known. But when the Receiver splits the received stream into words of length n, some of the words are not in C, due to bit-errors. Indeed, the received stream may contain any word z ∈ Bn . In order to understand the message, the Receiver must first decide which codeword c ∈ C was sent when z is received.

Definition 6.1 (Decision rule) Let C ⊆ Bn be a code. A decision rule for C is a function σ : Bn → C which assigns to each z ∈ Bn a codeword c ∈ C. Using a decision rule σ, the Receiver produces a final stream, and has effected a transformation T3:

received stream −→ final stream.

The entire system is illustrated in Figure 6.2.

original stream

T1 encoded coding stream X → C ⊆ Bn SENDER

T2 errors

CHANNEL

received stream

T3 decision rule Bn → C

final stream

RECEIVER

Figure 6.2 The three stages of the communication system

Definition 6.2 (Mistake) We say that a mistake occurs if a codeword in the final stream is not the same as the codeword in the corresponding position in the encoded stream. When a mistake occurs the Receiver will mis-interpret the message that was sent by the Sender. Mistakes are caused by the errors introduced at stage T2, transmission through a noisy channel. Our aim is to show that, by using an appropriate choice of code C at stage T1, and a suitable decision rule for C at stage T3, it is possible to make mistakes less likely.

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For example, consider the scenario described above, where the Sender is a controller emitting the symbols N,S,E,W, using the simple code N → 00, S → 01, E → 10, W → 11. The Receiver splits the received stream into words of length 2, and in this case all four possible words of length 2 are codewords (C = B2 ). Provided that the codewords are transmitted correctly, mistakes will not occur when the Receiver uses the obvious decision rule σ(z) = z

for all z ∈ B2 .

On the other hand, as we have already noted, if any bit is transmitted wrongly then a mistake is certain to occur. Fortunately it is possible to choose better codes.

Example 6.3 In the situation described above, suppose the Sender uses the code N → 000,

S → 110,

and the Receiver uses the decision rule

E → 101,

W → 011,

σ(000) = 000,

σ(100) = 011,

σ(010) = 000,

σ(001) = 101,

σ(110) = 110,

σ(101) = 101,

σ(011) = 011,

σ(111) = 110.

If a codeword is transmitted with no bit-errors, will a mistake occur? If the word 001 is received and it is assumed that one bit-error has been made, is it possible that a mistake will occur? Solution The proposed decision rule is such that, when z is codeword, σ(z) = z. Thus if a codeword is transmitted correctly, a mistake will not occur. What happens if one bit in a codeword is transmitted wrongly? Note that each codeword contains an even number of 1’s (either 0 or 2). Thus if there is one erroneous bit the received word will have an odd number of 1’s, and the error can be detected. However, if 001 is received (for example), then the Receiver must decide whether the transmitted codeword was 000, with an error in the last bit, or 101, with an error in the first bit, or 011, with an error in the second bit. The proposed decision rule assumes the second possibility, and so if 001 is received then there is a significant chance that a mistake will occur. In fact, there are other reasons why this rule is rather a poor one (see Section 6.3).

Example 6.4 In the same situation, suppose the Sender uses the code N → 000000,

S → 000111,

E → 111000,

W → 111111.

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93

The Receiver uses the decision rule that for any z, σ(z) = c is the codeword that is ‘most like’ z – that is, the codeword c for which z and c have most bits in common. In what circumstances will a mistake occur? Solution The codewords have been chosen so that if any one bit is changed, the resulting word is still ‘more like’ the original codeword than any other codeword. The proposed decision rule makes use of this property. For example, if 000000 is altered in transmission to 100000 then the Receiver will decide that 000000 was intended, because any other codeword would have to be affected by more than one bit-error to produce 100000. So an error in one bit is not only detected, but also corrected. For a mistake to occur, at least two bit-errors must be made. The system outlined above is a complicated one, involving several stages. In due course we shall be able to explain how the objective of making mistakes less likely can be achieved, but first we must study the various stages of the system in detail. • Section 6.2 quantifies the errors that occur when the bits in a codeword of length n are transmitted through a BSC with bit-error probability e. • Section 6.3 discusses the possible forms of a decision rule σ that assigns a codeword to each (possibly erroneous) received word. • Sections 6.4 and 6.5 deal with the correction of bit-errors, and how it depends on certain numerical parameters of the code that is used.

EXERCISES 6.1. In Example 6.3, suppose that 111 is received. If it is assumed that only one bit is in error, what are the possibilities for the intended instruction? 6.2. In Example 6.4 we proposed a decision rule σ based on the idea that σ(z) is the codeword that is ‘more like’ z than any other codeword. Using this rule, find the values of σ(z) for the following words z: 101000,

101111,

100111.

6.3. In the same scenario as in Exercise 6.2, suppose that the word 100100 is received. Is it possible to make a reasonable decision about which codeword was sent?

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6.2 The extended BSC Definition 6.5 (Product channel) Let Γ and Γ ′ be channels with input alphabets I, I ′ and output alphabets J, J ′ respectively. The product channel Γ ′′ = Γ × Γ ′ has input alphabet I × I ′ and output alphabet J × J ′ , and its matrix is given by the rule (Γ ′′ )ii′ jj′ = Γij Γi′′ j ′ . In other words Γ ′′ is a channel with inputs ii′ and outputs jj ′ , and Pr(output is jj ′ | input is ii′ ) = Pr(j | i)Pr(j ′ | i′ ). In the case Γ = Γ ′ , we denote Γ ×Γ ′ by Γ 2 . (Note that this does not correspond to the usual product of matrices.)

Example 6.6 Suppose Γ is the BSC with bit-error probability e. What is the channel matrix for Γ 2 ? Solution The inputs and outputs for Γ 2 are the four pairs 00, 01, 10, 11. If, for example, the input is 00 and the output is 10, this means that one bit has been transmitted wrongly (probability e), and one bit has been transmitted correctly (probability 1 − e). So the entry in the corresponding position in Γ 2 is e(1 − e). Using similar arguments, and labelling the rows and columns in the order 00, 01, 10, 11, the complete channel matrix is ⎞ ⎛ (1 − e)2 e(1 − e) e(1 − e) e2 2 2 ⎜ e(1 − e) (1 − e) e e(1 − e) ⎟ ⎜ ⎟. 2 2 ⎝ e(1 − e) (1 − e) e(1 − e) ⎠ e e2 e(1 − e) e(1 − e) (1 − e)2

Definition 6.7 (Extended channel) Suppose we are given a channel Γ with input set I and output set J, and a positive integer n. Then we define the extended channel Γ n to be the n-fold product of copies of Γ . The inputs to Γ n are words of length n in I, and the outputs are words of length n in J. If Γ is a binary symmetric channel, then we say that Γ n is an extended BSC.

6.2 The extended BSC

95

The inputs and outputs for an extended BSC are the members of the set Bn of all binary words of length n. We shall obtain a simple formula for the entries of the channel matrix Γ n , by generalizing the argument used above in the case n = 2. The following definition is crucial.

Definition 6.8 (Hamming distance) Suppose we are given two words x, y ∈ Bn , x = x1 x2 · · · xn

y = y1 y2 · · · yn .

The Hamming distance d(x, y) is the number of places where x and y differ; in other words, it is the number of i (1 ≤ i ≤ n) such that xi = yi . For example, consider the following words in B7 : x = 1010100,

y = 0110100,

z = 1011110.

The words x and y differ in the first and second bits only, so d(x, y) = 2. Similarly d(x, z) = 3 and d(y, z) = 4.

Theorem 6.9 Let x, y ∈ Bn . The entry (Γ n )xy in the channel matrix for the extended BSC with bit-error probability e is given by (Γ n )xy = ed (1 − e)n−d , where d = d(x, y).

Proof Suppose that x ∈ Bn is sent and y ∈ Bn is received. If d(x, y) = d, then d bits are in error, and the probability of this event is ed . The remaining n − d bits are correct, and the probability of this event is (1 − e)n−d . Hence (Γ n )xy = Pr(y | x) = ed (1 − e)n−d .

EXERCISES 6.4. Let Γ 2 be the channel matrix for the extended BSC with e = 0.01. Calculate the numbers in the row of Γ 2 corresponding to the input 00.

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6.5. Write down the channel matrix for Γ 2 when Γ is the binary asymmetric channel with parameters a, b (Exercise 5.1). 6.6. Let d denote the Hamming distance. Prove that, for all x, y, z ∈ Bn , d(x, x) = 0,

d(x, y) = d(y, x),

d(x, z) ≤ d(x, y) + d(y, z).

[The last property is known as the triangle inequality.]

6.3 Decision rules In Section 6.1 we introduced the idea of a decision rule σ for a code C ⊆ Bn . The idea is that when a word z ∈ Bn is received, the Receiver must try to make a reasonable choice as to which codeword σ(z) ∈ C was sent. As an example we considered the code C = {000, 110, 101, 011}, and the arbitrary decision rule σ given by σ(000) = 000,

σ(100) = 011,

σ(010) = 000,

σ(001) = 101,

σ(110) = 110,

σ(101) = 101,

σ(011) = 011,

σ(111) = 110.

Since the word 011 is actually a codeword, it is reasonable to define σ(011) = 011. On the other hand, 100 is not codeword, and putting σ(100) = 011 assumes that an error has been made in each of the three bits, which is clearly unreasonable. A good decision rule will depend on the communication system, in particular, the code C and the channel matrix Γ . The Receiver must use this information to formulate a decision rule that provides the best chance of making the right choice. At first sight, the obvious candidate is the following.

Definition 6.10 (Ideal observer rule) The ideal observer rule says that, given z, the Receiver should choose σ(z) to be a codeword c for which the probability that c was sent, given that z is received, is greatest. Unfortunately, the conditional probabilities occurring in this definition are of the form Pr(c | z), and these are not necessarily available to the Receiver.

6.3 Decision rules

97

In order to calculate Pr(c | z) it is necessary to equate two expressions for the probability tcz of an input-output pair (c, z): tcz = qz Pr(c | z)

tcz = pc Pr(z | c).

By definition Pr(z | c) is just the term Γcz in the channel matrix. However Pr(c | z) must be obtained from the equation Pr(c | z) = pc Γcz /qz , and this involves the input distribution p as well as Γ . In general the characteristics of the input will not be known to the Receiver – for example, when the original stream is generated by some remote-sensing device. However, it is reasonable to assume that the Receiver knows the characteristics of the channel, in particular the probabilities Γcz = Pr(z | c). Thus a more practical rule is the following.

Definition 6.11 (Maximum likelihood rule) The maximum likelihood rule says that, given z, the Receiver should choose σ(z) to be a codeword c that satisfies Pr(z | c) ≥ Pr(z | c′ ) for all c′ ∈ C. The maximum likelihood rule says that σ(z) should be a codeword c for which the probability that z is received, given that c is sent, is greatest. Writing the condition as Γcz ≥ Γc′ z we see that the rule can be expressed as follows. Given z, pick a largest term Γcz in column z of the channel matrix and put σ(z) = c. Note that there may be more than one c satisfying this condition, in which case the receiver must make an arbitrary choice. When the channel is the extended BSC we have a formula (Theorem 6.9) for the terms of the channel matrix, and this leads to a very simple interpretation of the maximum likelihood rule.

Definition 6.12 (Minimum distance rule) The minimum distance rule (or MD rule) says that given z ∈ Bn , the Receiver should choose σ(z) to be a codeword c such that d(z, c) is a minimum.

Theorem 6.13 For an extended BSC with e < to the MD rule.

1 2,

the maximum likelihood rule is equivalent

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Proof Suppose that c is codeword such that d(z, c) = i. Then, according to Theorem 6.9, Pr(z | c) = ei (1 − e)n−i . If c′ is codeword such that d(z, c′ ) = i′ , then  i′ −i Pr(z | c) 1−e ei (1 − e)n−i = i′ = . Pr(z | c′ ) e (1 − e)n−i′ e If e < 21 then (1 − e)/e > 1. Thus when i < i′ the right-hand side is greater than 1, and Pr(z | c) > Pr(z | c′ ). In other words, choosing c so that d(z, c) is minimal is the same as maximizing Pr(z | c). The theorem says that the Receiver can implement the maximum likelihood rule by choosing as σ(z) a codeword that is nearest to z, in the sense of Hamming distance. For some words z there will be more than one such codeword, and a choice will be needed. There are several ways in which the choice can be made. The simplest is for the Receiver to select one of the eligible codewords once and for all, and declare it to be σ(z). Alternatively the Receiver may choose one of the eligible codewords uniformly at random each time z occurs. In other words, if there are k eligible codewords, each one is chosen as σ(z) with probability 1/k. The second alternative seems more reasonable than the first, but from the theoretical point of view it creates some difficulties, because σ is no longer strictly a function (see Exercise 6.9). We shall continue to speak of ‘the MD rule’, remembering that in practice there may be more than one MD rule for any given situation.

Example 6.14 Suppose the code C ⊆ B8 has seven codewords c1 = 0000 0000 c2 = 0011 1000 c3 = 1100 0001 c4 = 0000 1110 c5 = 1011 1011 c6 = 0011 0110 c7 = 1100 1011. If σ is the MD rule, what value should the Receiver assign to (i) Solution

σ(1010 1011)

(ii) σ(1100 1001) ?

(i) The distances d(z, ci ) for z = 1010 1011 are i: 1 2 d(z, ci ) : 5 4

3 4 4 4

5 6 1 5

7 2

.

Hence the codeword nearest to z is c5 = 1011 1011. According to the theorem, this is also the codeword c for which the probability of receiving z, given that c was sent, is greatest.

6.3 Decision rules

99

(ii) The distances d(z ′ , ci ) for z ′ = 1100 1001 are i: 1 ′ d(z , ci ) : 4

2 3 5 1

4 5

5 6 4 8

7 1

.

In this case there are two codewords c3 and c7 that could be chosen as σ(z ′ ). The Receiver may either fix on one of them, or whenever z ′ occurs, choose one of them at random (say by tossing a fair coin).

EXERCISES 6.7. Let C ⊆ B8 be the code discussed in Example 6.14, with codewords c1 = 0000 0000 c2 = 0011 1000 c3 = 1100 0001 c4 = 0000 1110 c5 = 1011 1011 c6 = 0011 0110 c7 = 1100 1011. Using the MD rule σ, find σ(z) when z is 1000 1011,

1011 1010,

1100 0101.

6.8. Consider the code C consisting of the 10 words in B5 that have exactly two bits equal to 1. If one bit-error is made in transmitting the codeword 11000, what are the possible received words z? For each such z, make a list of the codewords that are nearest to z. 6.9. Under the same conditions as in the previous exercise, suppose the Receiver uses the MD rule, with the proviso that if there are k codewords that are nearest to a given word z, one of them is chosen as σ(z) with probability 1/k. Suppose the codeword 11000 is sent, and probability of a single bit-error in transmission is e. Show that there is probability 7e/12 that the Receiver will decide (wrongly) that the codeword sent was 10001. 6.10. Let Γ be a binary asymmetric channel with property that 0 is always transmitted correctly, but 1 is transmitted as 0 with probability f (0 < f < 1). Write down the entries (Γ 3 )cz of the extended channel matrix that correspond to codewords in the code C = {000, 111}. What is the maximum likelihood decision rule for C, using this channel? 6.11. Show that the maximum likelihood rule is not the same as the MD rule for the channnel considered in the previous exercise.

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6.4 Error correction The purpose of a decision rule is to ensure that, as far as possible, bit-errors are corrected. We shall focus on the MD rule for the extended BSC, where the effectiveness of the rule depends upon certain parameters of the code. The reader will find it helpful to envisage the following discussion in geometrical terms.

Definition 6.15 (Minimum distance) Let d(c, c′ ) denote the Hamming distance between codewords c and c′ in a code C ⊆ Bn (Definition 6.8). The minimum distance of C is δ = min′ d(c, c′ ). c=c

For example, suppose C ⊆ B6 has four codewords 000000, 111000, 001110, 110011. The distances between the codewords are as follows: d(000000, 111000) = 3,

d(000000, 001110) = 3,

d(000000, 110011) = 4,

d(111000, 001110) = 4,

d(111000, 110011) = 3,

d(001110, 110011) = 5,

and so the minimum distance is δ = 3.

Definition 6.16 (Neighbourhood) For any word x ∈ Bn and any r ≥ 0, the neighbourhood of x with radius r is the set Nr (x) = {y ∈ Bn | d(x, y) ≤ r}. Equivalently, the neighbourhood Nr (x) contains all the words that can be obtained from x by making not more than r bit-errors. For example, take x = 11010 ∈ B5 . Then the neighbourhood N1 (x) contains x itself and the five words obtained by making one error in x, that is N1 (x) = {11010, 01010, 10010, 11110, 11000, 11011}.

Lemma 6.17 If C is a code with δ ≥ 2r + 1, then for any codewords c, c′ ∈ C, the neighbourhoods Nr (c) and Nr (c′ ) are disjoint (see Figure 6.3).

6.4 Error correction

101

c r

≥ 2r + 1

c′ r

Figure 6.3 If δ ≥ 2r + 1 then Nr (c) and Nr (c′ ) are disjoint

Proof Suppose Nr (c) and Nr (c′ ) are not disjoint, so they have a common member z, say. Then by the triangle inequality (Exercise 6.6), d(c, c′ ) ≤ d(c, z) + d(z, c′ ) ≤ r + r = 2r. But this contradicts the assumption that δ ≥ 2r + 1, since d(c, c′ ) cannot be less than δ.

Theorem 6.18 Suppose the Sender uses a code C having minimum distance δ ≥ 2r + 1, and the Receiver uses the MD rule. Then, provided that not more than r bit-errors are made in transmitting any codeword, no mistakes will occur: every received word will be restored to the correct codeword.

Proof Let c be a codeword, and z the corresponding received word. If not more than r bit-errors are made in transmission, then z is in Nr (c). According to the lemma, z is not in Nr (c′ ) for any c′ = c. In other words, d(c, z) < d(c′ , z) for all c′ = c. This means that the MD rule will correctly assign c to z.

Definition 6.19 (Error-correcting code) A code C ⊆ Bn is an r-error-correcting code if δ ≥ 2r + 1. The definition is based on the implicit assumption that the MD rule is used. We sometimes say (even more vaguely) that a code with δ ≥ 2r + 1 ‘corrects r errors’. For example, if δ is at least 3, then C corrects 1 error.

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EXERCISES 6.12. Construct a 1-error correcting code C ⊆ B6 with |C| = 5. 6.13. For the code C constructed in the previous exercise, how many words cannot be corrected on the assumption that at most one bit-error has been made? 6.14. Is the code with |C| = 5 constructed in Exercise 6.12 the largest possible 1-error-correcting binary code with words of length 6?

6.5 The packing bound When we try to construct error-correcting codes by ensuring that δ has a given value, we run into a problem. In geometrical terms, it is intuitively clear that choosing a set of codewords C in Bn , such that the codewords are ‘far apart’ (in the sense of Hamming distance), means that the number of codewords |C| is constrained. The following theorem quantifies this remark.

Theorem 6.20 (The packing bound) If C ⊆ Bn is a code with δ ≥ 2r + 1, then      n n |C| 1 + n + + ··· + ≤ 2n . 2 r

Proof Given a codeword c, the words z such that d(c, z) = i are the words formed by altering any i of the n bits in c. Hence the number of such words is the binomial number ni . The number of z such that d(c, z) ≤ r is thus     n n 1+n+ + ···+ , 2 r and this is the size of the neighbourhood Nr (c) for each c ∈ C. Since δ ≥ 2r + 1, it follows from Lemma 6.17 that all the neighbourhoods Nr (c) (c ∈ C) are disjoint (Figure 6.4). There are |C| such neighbourhoods, and 2n words altogether, so the result follows.

6.5 The packing bound

103

Figure 6.4 If δ ≥ 2r + 1 all the neighbourhoods Nr (c) are disjoint

Example 6.21 Suppose we want to issue ID numbers, in the form of strings of n bits, to 100 students. The students are likely to make errors when using their ID’s, so it has been decided that we must use a 2-error-correcting code. What is the least value of n for which this may be possible? Solution

For r = 2 errors and |C| = 100 the packing bound is   n(n − 1) that is 100 1 + n + ≤ 2n , 2 50(n2 + n + 2) ≤ 2n .

By trial and error, the least value of n for which this inequality holds is n = 14. Of course, the problem of constructing a suitable set of 100 codewords, each of length 14, remains. Let us denote by A(n, δ) the largest value of |C| for which there is a code C ⊆ Bn with minimum distance δ. The packing bound gives an upper limit for A(n, δ) but there is no guarantee that a code of that size actually exists. Indeed the problem of evaluating A(n, δ) exactly is, in general, very difficult.

Example 6.22 What is the value of A(10, 7), the largest possible size of a code C ⊆ B10 with minimum distance δ = 7? Solution

Here we require δ = 2r + 1 with r = 3, so the packing bound is      10 10 |C| 1 + 10 + + ≤ 210 , 2 3

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6. The problem of reliable communication

that is, |C| ≤ 1024/176. Since |C| is an integer, it follows that A(10, 7) ≤ 5. In fact A(10, 7) = 2. To prove this, we can suppose without loss of generality that one codeword is 0000000000. Then any other codeword must have at least 7 ones. Now two words of length 10 with at least 7 ones must have at least 4 ones in the same positions. Hence if these words are distinct, their distance can be at most 6, contrary to the condition δ = 7. It follows that there can be only one other codeword besides 0000000000. There is another, very useful, way of interpreting the constraint imposed by the packing bound. When C ⊆ Bn and |C| = m, a codeword c ∈ C conveys (at most) log2 m bits of information, but requires n bits of data. Hence the following definition.

Definition 6.23 (Information rate) The information rate of a code C ⊆ Bn is ρ=

log2 |C| . n

For example, suppose C ⊆ B6 has four codewords 000000, 111000, 001110, 110011. Here n = 6 and |C| = 4 = 22 , so the information rate is ρ = 2/6 = 1/3. This corresponds to the fact that C requires 6 bits of data to encode 2 bits of information. If we suppose that one megabyte of data can be transmitted per second then, when C is used to code the data, only one-third of a megabyte of information is actually transmitted per second. Constructing a code C ⊆ Bn with given size |C| and a given value of δ is equivalent to constructing a code with given values of n, ρ, and δ. The packing bound tells us that there is a ‘trade-off’ between δ and ρ.

EXERCISES 6.15. If we require a code C ⊆ B6 with information rate at least 0.35, what is the smallest possible value of |C|? 6.16. Show that (log2 9)/6 ≈ 0.528 is an upper bound for the information rate of a 1-error-correcting code C ⊆ B6 .

6.5 The packing bound

105

6.17. Suppose we want to issue ID numbers, in the form of strings of n bits, to 1000 people, using a 1-error-correcting code. What is the least value of n for which this may be possible? 6.18. Using a general form of the proof that A(10, 7) = 2 (Example 6.22), show that if δ > 2n/3 then A(n, δ) = 2. 6.19. Prove that there is an r-error-correcting code C ⊆ Bn satisfying      n n |C| 1 + n + + ···+ ≥ 2n . 2 2r [This result gives a lower bound for A(n, 2r + 1). It is known as the Gilbert-Varshamov bound.] 6.20. An r-error-correcting code C ⊆ Bn is said to be perfect if the neighbourhoods Nr (c) exactly cover Bn – that is, every word in Bn is in precisely one Nr (c). Show that if r = 1 and such a code exists, then n must be of the form 2m − 1. [See also Section 9.1.]

Further reading for Chapter 6 There are many books on error-correcting codes, at various levels. The standard works by MacWilliams and Sloane [6.2], and Pless and Huffman [6.3] are recommended for a thorough treatment of the subject. A readable account of the early history, and the connections with geometry and algebra, is given by Thompson [6.4]. Estimates of the numbers A(n, δ) are constantly being improved. A up-todate list is maintained by Litsyn, Rains, and Sloane [6.1]. 6.1 S. Litsyn, E.M. Rains, N.J.A. Sloane. Table of Nonlinear Binary Codes. http://www.research.att.com/∼njas/codes/And/ 6.2 F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977). 6.3 V. Pless and W. Huffman (eds.). Handbook of Coding Theory (2 vols). Elsevier, Amsterdam (1998). 6.4 T.M. Thompson. From Error-Correcting Codes through Sphere Packings to Simple Groups. Carus Mathematical Monographs 21, Math. Assoc. of America (1983).

7 The noisy coding theorems

7.1 The probability of a mistake Here again (Figure 7.1) is the model of communication outlined at the start of the previous chapter.

original stream

T1 encoded coding stream X → C ⊆ Bn SENDER

T2 errors

CHANNEL

received stream

T3 decision rule Bn → C

final stream

RECEIVER

Figure 7.1 A model of a communication system

Suppose that the original stream is emitted by a source that produces symbols from an alphabet X, according to the probability distribution p on X. When the Sender uses the code C, the probability that a codeword c ∈ C is sent is the same as the probability that the source emits the symbol x ∈ X for which c is the codeword. Thus we can regard the encoded stream as being emitted by a source (C, p). Since p may be unknown, the appropriate decision rule is the maximum likelihood rule. Given that the channel is the extended BSC, the Receiver can use its equivalent form, the MD rule. N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 7, 

108

7. The noisy coding theorems

For each c ∈ C, denote by F (c) the set of words to which the MD rule σ does not assign c: F (c) = {z ∈ Bn | σ(z) = c}. According to Definition 6.2 a mistake occurs when a codeword c ∈ C is altered in transmission and the received word z is in F (c). In this situation, the Receiver will decide that a codeword c′ different from c was sent. For a given codeword c let Mc denote the probability that when c is sent the received word z is in F (c), in other words Mc is the sum of the conditional probabilities Pr(z | c) for z ∈ F (c):  Pr(z | c). Mc = z∈F (c)

Definition 7.1 (Probability of a mistake) The probability of a mistake when the encoded stream corresponds to a source (C, p) is  pc M c , M (C, p) = c∈C

where Mc is given by the formula displayed above. The aim is to choose the code C so that M (C, p) is as small as we please. The following example illustrates how this might be achieved.

Example 7.2 An investor wishes to communicate the messages BU Y , SELL to her stockbroker. Suppose she uses one of the codes (i) BU Y → 0,

SELL → 1;

(ii) BU Y → 000,

SELL → 111.

The encoded messages are transmitted via an extended BSC with bit-error probability e, and the stockbroker uses the MD rule. Which code has the smaller probability of a mistake? Solution (i) Here n = 1 and the code is C1 = {0, 1}. The MD rule is σ(0) = 0, σ(1) = 1, and the sets F (c) are F (0) = {1},

F (1) = {0}.

The probabilities M0 and M1 are M0 = Pr(1 | 0) = e,

M1 = Pr(0 | 1) = e.

7.1 The probability of a mistake

109

Hence, for any distribution p = [p, 1 − p] on C1 , M (C1 , p) = pe + (1 − p)e = e. (ii)

Here n = 3 and the code is C3 = {000, 111}. The MD rule is σ(000) = σ(100) = σ(010) = σ(001) = 000, σ(011) = σ(101) = σ(110) = σ(111) = 111.

With this rule F (000) = {011, 101, 110, 111},

F (111) = {000, 001, 010, 100}.

If the codeword 000 is sent, the probability M000 of a mistake is thus Pr(011 | 000) + Pr(101 | 000) + Pr(110 | 000) + Pr(111 | 000) = e2 (1 − e) + e2 (1 − e) + e2 (1 − e) + e3 . A very rough approximation tells us that all four terms are less that e2 , so M000 < 4e2 . A similar calculation shows that M111 < 4e2 also. Hence, for any distribution p = [p, 1 − p] on C3 , M (C3 , p) < p(4e2 ) + (1 − p)(4e2 ) = 4e2 . Thus, when e is small, M (C3 , p) is much less than M (C1 , p). For example, when e = 0.001, we have M (C1 , p) = 0.001,

M (C3 , p) < 0.000004.

In general, can we give estimates for the probability of a mistake, based on arguments like those used in the example? The following simple remark is a starting point.

Lemma 7.3 If C ⊆ Bn is an r-error-correcting code, and z is in F (c), then d(z, c) ≥ r + 1.

Proof Theorem 6.18 says that if the received word z is such that d(z, c) ≤ r, then the Receiver will correctly decide that c was sent. Hence the words for which the Receiver decides on the wrong codeword must satisfy d(z, c) ≥ r + 1.

110

7. The noisy coding theorems

When a code C ⊆ Bn is used we have (Theorem 6.9) Pr(z | c) = (Γ n )cz = ed(c,z) (1 − e)n−d(c,z) . Hence Mc =



z∈F (c)

ed(c,z)(1 − e)n−d(c,z) .

According to Lemma 7.3 all the summands in Mc are of the form ei (1 − e)n−i ,

for some i ≥ r + 1.

Since 0 < e < 1, we have ei (1 − e)n−i < ei ≤ er+1 . In other words, Mc is a sum of terms of order er+1 . In specific cases we can estimate the size of F (c), and hence obtain an upper bound for Mc . Thus, in Example 7.2, C3 is a 1-error correcting code, and so each term in Mc is of order e2 . There are four words in F (c), so we can conclude that Mc is bounded above by 4e2 . The problem with this approach is that F (c) is not necessarily a small set. This means that the suggested upper bound  Mc ≤ er+1 = |F (c)| er+1 , z∈F (c)

is such that the ‘constant’ multiplier of er+1 may be very large. For this reason a more careful analysis of the problem is needed.

EXERCISES 7.1. Let C = {00000, 11100, 00111}. If σ is an MD decision rule for C, show that F (c) is a set of size 12, at least, for each codeword c. 7.2. Consider the code C = {0, 1} as the input to a BSC with bit-error probability e = 0.2. Verify that the maximum likelihood decision rule is given by σ(0) = 0, σ(1) = 1, and that the probability of a mistake is 0.2, for any distribution p on C. 7.3. In the previous exercise, suppose the input distribution is p = [0.9, 0.1], and this is known to the Receiver, who can therefore use the ideal observer rule (Definition 6.10). Write down the rule explicitly, and show that using this rule the probability of a mistake is reduced to 0.1.

7.2 Coding at a given rate

111

7.4. The repetition code Rn ⊆ Bn consists of the two words 000 · · · 00 and 111 · · · 11. If n is an odd number 2ℓ + 1, and σ is the MD rule, find the size of the sets F (000 · · · 00) and F (111 · · · 11). 7.5. Suppose that R5 is transmitted through an extended BSC with biterror probability e, and the MD rule is used. Calculate M00000 , M11111 , and M (R5 , p) exactly as functions of e, for any distribution p on R5 . 7.6. Show that, provided e < 41 , we can make M (R2ℓ+1 , p) as small as we please by choosing ℓ sufficiently large. 7.7. In Exercise 6.10 we considered a binary asymmetric channel Γ with the property that 0 is always transmitted correctly, but 1 is transmitted as 0 with probability f (0 < f < 1). Suppose the source (Rn , p) is transmitted through the extended channel Γ n , where p = [p, 1−p]. What is the behaviour of the information rate and the probability of a mistake as n → ∞?

7.2 Coding at a given rate We continue to discuss the model shown in Figure 7.1. Specifically, stage T1 represents encoding with a binary code C, stage T2 represents transmission through an extended BSC, and stage T3 represents the application of the MD rule. Question: Is it possible to choose the code C so that the probability of a mistake is arbitrarily small, while information is transmitted at a given rate ρ? We shall see that the answer is yes, provided ρ is not too large. In fact, the key to maintaining a given information rate ρ is to code blocks of symbols, rather than individual symbols. It is convenient to suppose that the original stream is already in the form of a string of bits. This can be arranged by using a simple ‘pre-coding’ rule. For example, if the original stream is a sequence of commands N, S, E, W (as in Section 6.1), we could use the obvious pre-coding N → 00, S → 01, E → 10, W → 11. Consider the following strategy for stage T1. The Sender divides the original stream of bits into blocks of a certain size, say k, and assigns to each block a codeword belonging to a code C. There are 2k possible blocks, so a code of size |C| = 2k will be required. In order to ensure that the information rate is not

112

7. The noisy coding theorems

less than some given value ρ, the Sender must determine the appropriate length n for the codewords. The information rate of the code C is (log2 |C|)/n = k/n. Hence the Sender must choose the parameters k, n, and the code C ⊆ Bn , such that |C| = 2k and k ≥ ρn.

The Sender also wishes to choose the code C so that the probability of a mistake is small, knowing that the encoded stream will be transmitted through an extended BSC with bit-error probability e, and the Receiver will use the MD decision rule. These conditions imply that the trade-off between the parameters ρ and δ will operate.

Example 7.4 Suppose that the desired information rate is ρ = 0.8, and the Sender wishes to use a 1-error correcting code (δ ≥ 3). What are the smallest possible values of n and k? Solution If C ⊆ Bn is a 1-error correcting code with |C| = 2k , it follows from the packing bound (Theorem 6.20) that 2k (1 + n) ≤ 2n . Also, in order to achieve the rate ρ = 0.8, the Sender must ensure that k is not less than 0.8n. Thus we we require the least value of n for which there is an integer k satisfying n + 1 ≤ 2n−k

and

5k ≥ 4n.

It is easy to check (by trial and error) that the smallest possible solution is n = 25, k = 20. The example shows that at least 220 (about one million) codewords of length 25 are needed to achieve the required rate ρ = 0.8. In Section 8.4 we shall give a simple method for constructing a suitable code with these parameters, but of course it is too large to be written down explicitly. For smaller rates, such as ρ = 0.5, we can give simpler examples.

Example 7.5 Let C be the code that assigns to each block of three bits, say y1 y2 y3 , the codeword x1 x2 x3 x4 x5 x6 ∈ B6 defined as follows: x1 x2 x3 x4 x5 x6

= y1 = y2 = y3 =0 =0 =0

if y1 = y2 , if y2 = y3 , if y1 = y3 ,

x4 = 1 otherwise, x5 = 1 otherwise, x6 = 1 otherwise.

7.3 Transmission using the extended BSC

113

Show that C is a 1-error-correcting code, and its rate is ρ = 0.5. Solution

Explicitly the code is

000 → 000000,

001 → 001011,

010 → 010110,

100 → 100101,

011 → 011101,

101 → 101110,

110 → 110011,

111 → 111000.

Checking the distances between pairs of codewords, it turns out that the minimum distance is 3. (In Chapter 8 we shall describe a quicker method of verifying this fact.) Thus C is indeed a 1-error correcting code, and since n = 6 and k = 3 the information rate is 1/2. In the rest of this chapter we shall study the general problem illustrated in the preceding examples. As well as the problem of choosing a suitable code C, we must consider how the situation is affected by changes in the source distribution p. The aim, of course, is to make M (C, p) as small as we please, for any p.

EXERCISES 7.8. Suppose the Sender wishes to code blocks of size k with words of length n, using a 1-error correcting code. If it is required to transmit with information rate not less than 0.6, what are the smallest possible values of n and k? 7.9. In Example 7.5 there is a code for blocks of size 3 using words of length 6. An alternative would be to code each block by repeating it twice: for example, 101 → 101101. Compare the parameters of the two codes. 7.10. Is it possible to carry out a construction similar to that in Example 7.5, so that blocks of size four are represented by codewords of length seven, and the minimum distance is 3? [In Chapter 8 we consider this problem in more general terms.]

7.3 Transmission using the extended BSC In this section we consider stage T2 of our model, transmission of the encoded stream using an extended BSC, with the possibility that bit-errors may occur. We begin by showing that if the capacity of the BSC Γ is γ, then the capacity of Γ n is nγ. Intuitively, this is because Γ n can be regarded as n copies

114

7. The noisy coding theorems

of Γ ‘in parallel’, and the copies are independent. The encoded stream may have complex interdependencies among its bits, but when each individual bit is transmitted through Γ n there is the same probability that an error will occur, irrespective of what happens to the other bits. This fact is implicit in the general definition of Γ n as the product of n copies of Γ (Definitions 6.6 and 6.7). In our model the inputs to Γ n are codewords c1 . . . cn ∈ C and the outputs are words z1 . . . zn ∈ Bn . We have (Γ n )c1 ...cn

z1 ...zn

= Pr(z1 . . . zn | c1 . . . cn ) = Pr(z1 | c1 ) · · · Pr(zn | cn ) = Γc1 z1 · · · Γcn zn .

For the BSC we have an explicit formula for the entries of Γ n , but it is not needed here. As in Section 7.1, we suppose that the encoded stream is emitted by a source (C, p). Then the received stream is a source (Bn , q), where q = pΓ n . We shall calculate the capacity of Γ n by the method described in Section 5.5.

Lemma 7.6 Let Γ be the BSC with bit-error probability e. Then, with the notation as above, H(q | p) = nh(e).

Proof In Section 5.5 we defined the uncertainty of the output for a given input c1 . . . cn as  1 H(q | c1 . . . cn ) = (Γ n )c1 ...cn z1 ...zn log n . (Γ )c1 ...cn z1 ...zn z ...z n

1

n

Using the definition of Γ , this can be written as 

z1 ...zn

=

Γc1 z1 · · · Γcn zn log  z1

Now



z

...

 zn

1 Γc1 z1 · · · Γcn zn

Γc1 z1 · · · Γcn zn

n  i=1

log

1 . Γci zi

Γcz = 1 for each c, so the expression reduces to n   i=1 zi

Γci zi log

1 Γci zi

.

7.3 Transmission using the extended BSC

115

The values of zi are 0 and 1, so each sum over zi is simply Γci 0 log

1 1 + Γci 1 log . Γci 0 Γci 1

One summand is e log(1/e) and the other is (1 − e) log(1/(1 − e)). Hence the sum is h(e), and it follows that H(q | c1 . . . cn ) = nh(e) for all c1 . . . cn . It follows from Lemma 5.13 that  p(c1 . . . cn ) H(q | c1 . . . cn ) = nh(e). H(q | p) = c1 ...cn

Theorem 7.7 If the capacity of the BSC Γ is γ = 1 − h(e), then the capacity of Γ n is nγ.

Proof We use the fact (Section 5.5) that the capacity is the maximum value of H(q) − H(q | p). Lemma 7.6 shows that in this case H(q | p) is a constant, nh(e), so we have only to maximize H(q). Since q is a distribution on the set Bn of size 2n , it follows from Theorem 3.11 that H(q) ≤ log2 (2n ) = n, and the bound is attained if and only if all elements of Bn are equally probable. Hence H(q) − H(q | p) ≤ n − nh(e) = nγ, and the bound is attained. We now consider the uncertainty of the situation from the Receiver’s viewpoint. In Chapter 5 we denoted this quantity by H(Γ n ; p) = H(p | q). In this case it represents the uncertainty (per codeword) of the encoded stream (C, p), given the received stream. Since each codeword has n bits, the uncertainty per bit is H(Γ n ; p)/n. The following theorem shows that, when the required information rate exceeds the capacity of the channel, this quantity cannot be made arbitrarily small for all distributions p, however C and n are chosen.

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7. The noisy coding theorems

Theorem 7.8 Let C ⊆ Bn be a code with information rate ρ, and let p∗ be the distribution in which each codeword in C is equally probable. Suppose that the stream emitted by the source (C, p∗ ) is transmitted through the extended BSC Γ n , where Γ has capacity γ. Then H(Γ n ; p∗ ) ≥ n(ρ − γ).

Proof By definition, the capacity of Γ n is the maximum of H(p) − H(Γ n ; p), taken over all distributions p. In particular, when p = p∗ , H(p∗ ) − H(Γ n ; p∗ ) ≤ nγ. Now H(p∗ ) = log |C| (Theorem 3.11), so we have

H(Γ n ; p∗ ) ≥ log |C| − nγ.

Since the information rate of C is ρ = log |C|/n, the inequality can be written in the form stated above. This result is trivial when ρ < γ, since the right-hand side is negative and it is always true (by definition) that the left-hand side is non-negative. However, as we shall see, the result is highly significant when ρ > γ.

EXERCISES 7.11. Suppose that a binary code with information rate 0.9 is being transmitted through an extended BSC with bit-error probability 0.03. Does the rate exceed the capacity? If each codeword is equally probable, what can be said about the uncertainty from the viewpoint of the Receiver? 7.12. Let Γ be an arbitrary channel. Denote its 2-fold extension by Γ 2 , and the input distribution by p, so that the output distribution is q = pΓ 2 . Use the method given in Lemma 7.6 to show that H(q | p) = H(q′ | p′ ) + H(q′′ | p′′ ), where p′ and p′′ are the marginal distributions associated with p, and q′ = p′ Γ , q′′ = p′′ Γ . 7.13. Deduce from the previous result that if the capacity of Γ is γ then the capacity of Γ 2 is 2γ.

7.4 The rate should not exceed the capacity

117

7.4 The rate should not exceed the capacity In this section we shall prove that if the probability of a mistake is required to be arbitrarily small, then the capacity of the channel is an upper bound to the rate at which information can be transmitted. Consider stage T3 of our model, where the Receiver converts the received stream into a final stream using the MD rule. The final stream is a sequence of codewords, purporting to have been produced by the source (C, p), and there are two elements of uncertainty associated with it. The first element of uncertainty arises from the fact that the Receiver does not know whether a mistake has been made. The probability of a mistake is M = M (C, p), and the probability of no mistake is 1 − M . The uncertainty associated with this situation is h(M ), where as usual h(M ) = M log(1/M ) + (1 − M ) log(1/(1 − M )). The second element of uncertainty rises from the fact that, if a mistake has been made, then the correct codeword has been replaced by an incorrect one. But in this case the Receiver does not know which of the |C| − 1 other codewords is the correct one. The probability of a mistake is M , and the uncertainty associated with |C| − 1 choices is at most log(|C| − 1), so this makes a contribution of at most M log(|C| − 1) to the total uncertainty. The application of the decision rule can only increase the uncertainty. Thus the uncertainty of the final stream, which is the sum of the two quantities described above, is an upper bound for the uncertainty of the received stream: H(Γ n ; p) ≤ h(M ) + M log(|C| − 1). This result is known as Fano’s inequality, and it will play an important part in our next theorem. A formal proof is given in Section 7.6. Recall that the aim is to encode a stream of bits in such a way that •1 •2

information is transmitted at a given rate ρ; the probability of a mistake, M , is arbitrarily small.

In order to satisfy these criteria we must construct codes Cn ⊆ Bn for a infinite sequence of values of n. If |Cn | = 2kn , criterion •1 will be satisfied provided that kn is at least ρn. If that is so, we can split the stream into blocks of size kn and assign a codeword in Cn to each block. However, the next result shows that we cannot also satisfy •2 if ρ is greater than γ.

118

7. The noisy coding theorems

Theorem 7.9 Suppose that, for an infinite sequence of values of n, we have constructed codes Cn ⊆ Bn such that |Cn | ≥ 2ρn . Let p∗ be the equiprobable distribution on Cn . If ρ > γ, then the probability of a mistake M (Cn , p∗ ) does not tend to zero as n → ∞.

Proof Let |Cn | = 2kn where kn ≥ ρn, and Mn = M (Cn , p∗ ). Fano’s inequality says that H(Γ n ; p) ≤ h(Mn ) + Mn log(|Cn | − 1). Since h(Mn ) ≤ 1 and log(|Cn | − 1) < log |Cn | = kn , it follows that, for any p, H(Γ n ; p) < 1 + Mn kn . On the other hand, it follows from the proof of Theorem 7.8 that H(Γ n ; p∗ ) ≥ log |Cn | − nγ = kn − nγ. Combining these inequalities we get 1 + Mn kn > kn − nγ, and so Mn > 1 −

nγ + 1 nγ + 1 . ≥1− kn nρ

As n → ∞ the last expression approaches 1 − γ/ρ, and since ρ > γ this is strictly positive. Hence the limit of Mn (if it exists) is not zero.

EXERCISES 7.14. Suppose that the extended BSC is being used to transmit codewords of length 18, and the MD rule is being used by the Receiver. It is found experimentally that a code with about 64000 codewords can be transmitted with negligible probability of a mistake. What conclusion can be drawn about the bit-error probability e? 7.15. Investigate Fano’s inequality in the case when n = 1, C = {0, 1}, and Γ is the BSC with e = 0.5. What is special about this case?

7.5 Shannon’s theorem

119

7.5 Shannon’s theorem The most striking theoretical result in Information Theory is Shannon’s Theorem, which was also historically the first result in the area. It is logically equivalent to the converse of Theorem 7.9. Thus it asserts that if ρ < γ then it is possible to find an infinite sequence of codes Cn such that Cn ⊆ Bn ,

|Cn | ≥ 2ρn ,

and M (Cn , p) → 0 as n → ∞.

For example, suppose we wish to transmit a stream of bits, using a device for which the bit-error probability is estimated to be 0.03. It is specified that the probability of a mistake must be less than 10−6 . Knowing Shannon’s Theorem, we could try to design our system in the following way. • Step 1 Choose a specific value of ρ less than γ. Here the capacity of the BSC with e = 0.03 is γ = 1 − h(e) = 0.8 approximately, so a suitable value is ρ = 0.75. • Step 2 that

On the basis of Shannon’s Theorem, choose a code Cn ⊆ Bn such |Cn | = 2k (where k ≥ 0.75n),

and M (Cn , p) < 10−6 .

• Step 3 Divide the original stream into blocks of length k and encode the blocks using codewords in Cn . • Step 4 Transmit the encoded stream using the given device, and apply the MD rule to the received stream. In practice, we should soon run into difficulties if we tried to implement this plan. At Step 2 Shannon’s Theorem tells us that a suitable code Cn exists, but it does not tell us how to find it. The same problem occurs in Step 3, where we have to set up a rule that assigns a codeword of length n to each block of k bits: in other words, we have to specify an encoding function Bk → Bn . Thus, although Shannon’s Theorem is a seminal result, it is not a set of instructions for practitioners. The proof is very ingenious, but it does not form the basis of a method for constructing suitable codes, and for that reason we shall not go into the details. Instead we shall move on (in Chapter 8) to describe some of the simple mathematical techniques that can be used to construct good encoding functions Bk → Bn for many values of ρ = k/n. Fano’s inequality has shown that the attempt is pointless if ρ > γ, while Shannon’s Theorem guarantees that it is worthwhile if ρ < γ.

120

7. The noisy coding theorems

EXERCISES 7.16. Suppose we are using a BSC with known bit-error probability 0.05. How should we go about choosing values of k and n so that Shannon’s theorem will guarantee the existence of codes with arbitrarily small values of M ?

7.6 Proof of Fano’s inequality The proof of Fano’s inequality is elementary, but rather long. We begin with some notation and a couple of lemmas. For each z ∈ Bn , the codeword σ(z) ∈ C will be denoted by z ∗ . With this notation, given a codeword c ∈ C the set F (c) defined in Section 7.1 is the set of z such that z ∗ = c. As a temporary notation, let K be the set of pairs (c, z) with this property: K = {(c, z) ∈ C × Bn | c = z ∗ }. Recall that the joint probability distribution t on C × Bn is such that tcz = pc Pr(z | c) = pc (Γ n )cz .

Lemma 7.10 Let M = M (C, p) be the probability of a mistake when the source (C, p) is transmitted through the extended BSC Γ n and the MD decision rule is used. Then   tz ∗ z . M= tcz and 1 − M = z

(c,z)∈K

Proof According to Definition 7.1 the probability of a mistake is    (Γ n )cz = pc pc M c = M = M (C, p) = c

c

z∈F (c)



(c,z)∈K

Hence 1−M =



(z,c)∈K /

tcz =

 z

c=z ∗

tcz =

 z

tz ∗ z .

tcz .

7.6 Proof of Fano’s inequality

121

Lemma 7.11 If the source (C, p) is transmitted though the extended BSC Γ n and the probability of receiving z is qz , then   H(Γ n ; p) = tcz log(qz /tcz ) + tz∗ z log(qz /tz∗ z ). z

(c,z)∈K

Proof By definition, H(Γ n ; p) = H(t) − H(q), where q = pΓ n and   tcz log(1/tcz ), H(q) = qz log(1/qz ). H(t) = z

(c,z)



it follows that   H(t) − H(q) = tcz log(qz /tcz ) = tcz log(qz /tcz ).

Since qz =

c tcz

z

(c,z)

c

For each z, the sum over c contains one term with c = z ∗ . Separating this term from the rest, we have   H(Γ n ; p) = tcz log(qz /tcz ) + tz∗ z log(qz /tz∗ z ). z

z

c=z ∗

Writing the double sum as a sum over the set K, the result follows.

Theorem 7.12 (Fano’s inequality) Let C ⊆ Bn , and let M = M (C, p) be the probability of a mistake when the source (C, p) is transmitted through the extended BSC Γ n , and the MD decision rule is used. Then H(Γ n ; p) ≤ h(M ) + M log(|C| − 1).

Proof Using the expressions for M and 1 − M obtained in Lemma 7.10 we can write h(M ) as the sum of two terms   h(M ) = tcz log(1/M ) + tz∗ z log(1/(1 − M )). (c,z)∈K

z

Lemma 7.11 gives a similar expression for H(Γ n ; p). Combining these expressions we have H(Γ n ; p) − h(M ) = S1 + S2 ,

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where S1 =



tcz log(M qz /tcz ),

S2 =

 z

(c,z)∈K

tz∗ z log((1 − M )qz /tz∗ z ).

We shall use the Comparison Theorem 3.10 to prove that S1 ≤ M log(|C| − 1),

S2 ≤ 0.

For S1 , it is easy to verify that both vcz = tcz /M and wcz = qz /(|C| − 1) are probability distributions on the set K, so that   0≥ vcz log(wcz /vcz ) = (tcz /M ) log(M qz /((|C| − 1)tcz ) K

=



K

(tcz /M ) log(M/tcz ) +

K

 K

(tcz /M ) log(qz /(|C| − 1))

= M −1 S1 + log(1/(|C| − 1)). Thus S1 ≤ M log(|C| − 1). Similarly, for S2 both qz and uz = tz∗ z /(1 − M ) are probability distributions on the set Bn , so that   0≥ uz log(qz /uz ) = (tz∗ z /(1 − M )) log((1 − M )qz /tz∗ z ) = (1 − M )−1S2 . z

z

Thus S2 ≤ 0. Putting the two bounds together we have the result.

Further reading for Chapter 7 Fano’s inequality is one of the many contributions to information theory that appear in his book [7.1]. Proofs of Shannon’s Theorem in the form described in Section 7.5 can be found in Welsh’s book [3.5], and in many other texts. The theorem can be stated and proved in a much more general way, as described in the books by Ash [5.1] and McEliece [5.2]. 7.1 R.M. Fano. Transmission of Information: A Statistical Theory of Communications. MIT Press, Cambridge, Mass. (1961).

8 Linear codes

8.1 Introduction to linear codes The techniques for constructing useful codes can be extended enormously if we endow the symbols with number-like properties. The ‘arithmetic’ codes for data compression described in Chapter 4 are an example. Now we shall use ‘algebraic’ methods in order to construct codes for the purpose of error-correction. In the case of the binary alphabet B, the symbols 0 and 1 can be ‘added’ and ‘multiplied’ according to the rules 0 + 0 = 0,

0 + 1 = 1,

1 + 0 = 1,

1 + 1 = 0;

0 × 0 = 0,

0 × 1 = 0,

1 × 0 = 0,

1 × 1 = 1.

In the language of Abstract Algebra, we say that the set B, with these operations, is a field. We shall emphasize the algebraic structure by using the notation F2 for this field. The most useful method of constructing binary codes depends on the fact that the set F2 n of words of length n in F2 is a vector space. The rules for vector addition and scalar multiplication are the obvious ones: (x1 x2 . . . xn ) + (y1 y2 . . . yn ) = (x1 + y1 x2 + y2 . . . xn + yn ); 0(x1 x2 . . . xn ) = (00 . . . 0),

1(x1 x2 . . . xn ) = (x1 x2 . . . xn ).

Recall that a subset C of the vector space F2 n is a subspace if whenever x, y ∈ C it follows that x + y ∈ C.

N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 8, 

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8. Linear codes

Definition 8.1 (Linear code) A (binary) linear code is a subspace C of the vector space F2 n . For example, the subset C = {000, 110, 011} of F2 3 is not a linear code, because 110 + 011 = 101, which is not in C. On the other hand, the repetition code Rn ⊆ F2 n , containing the two words 000 · · · 00 and 111 · · · 11 is a linear code for any n, since 111 · · · 11 + 111 · · · 11 = 000 · · · 00. Note that every linear code must contain the all-zero word 000 · · · 00, since x + x = 000 · · · 00 for any x ∈ F2 n . We have seen that the construction of good codes involves a trade-off between the information rate ρ and the minimum distance δ. In the case of linear code we can be more specific about these parameters. Since a linear code C is a subspace of F2 n it has a dimension k, defined as the size of a minimal spanning set (basis). Every element of C can be expressed uniquely as a linear combination of the basis, and so |C| = 2k for some k in the range 0 ≤ k ≤ n. It follows that the information rate of C is ρ=

k log2 |C| = . n n

Thus it is convenient to use k instead of ρ, and we usually describe a linear code by the parameters (n, k, δ). For codes in general, finding δ is tedious, because it requires the comparison of every pair of codewords, but for a linear code there is a relatively easy way.

Definition 8.2 (Weight) The weight w(x) of a word x ∈ F2 n is the number of 1’s in x. Equivalently, w(x) = d(x, 0), where 0 denotes the all-zero word 000 · · · 00.

Lemma 8.3 For a linear code, the minimum distance δ is equal to the minimum weight of a nonzero codeword.

Proof Suppose c1 and c2 are codewords such that c1 = c2 . A bit in c1 + c2 is 1 if and only if the corresponding bits in c1 and c2 are 0 and 1 in some order – that is, they are different. It follows from the definition of d that d(c1 , c2 ) = w(c1 + c2 ).

8.1 Introduction to linear codes

125

Since C is linear, c1 + c2 is also in C. Hence each value of d(c1 , c2 ) is the weight of a codeword, and the minimum value is the minimum weight.

Example 8.4 Find the parameters (n, k, δ) of the following linear codes. D1 = {000000, 100000, 010000, 110000}, D2 = {000000, 111000, 000111, 111111}. Solution D1 has words of length n = 6 and since there are 4 = 22 codewords the dimension is k = 2 (and the rate is ρ = 1/3). The weights of the nonzero codewords are is 1, 1, 2, so δ = 1. With this code, no error correction is possible. D2 also has dimension 2 and rate 1/3, but the minimum distance is δ = 3, and it is a 1-error-correcting code. The construction of linear codes C with given values of the parameters (n, k, δ) is constrained by the packing bound, Theorem 6.20. When |C| = 2k , and δ ≥ 2r + 1 (so that C is an r-error-correcting code) the condition is        n n n 2k 1 + + + ··· ≤ 2n . 1 2 r

Example 8.5 Find an upper bound for the number of codewords in a linear code with word length 12, if a 2-error-correcting code is required. Solution The packing bound is 2k (1 + 12 + 66) ≤ 4096, so 2k ≤ 50 approximately. Since k must be a integer the maximum possible value of 2k is 25 = 32. (Note that, as yet, we have no means of constructing such a code.)

EXERCISES 8.1. Which of the following subsets of F2 3 is a linear code? C1 = {000, 110, 101, 011}, C2 = {000, 100, 010, 001}. 8.2. Suppose that we wish to send any one of 128 different messages, and each message is to be represented by a binary codeword of length 10. Is it possible to construct a 1-error-correcting linear code satisfying these conditions?

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8. Linear codes

8.3. Professor MacBrain has decided that each student in his department will be allotted an identity number in the form of a binary word. (a) If there are 53 students, find the least possible dimension of a linear code for this purpose. (b) If the code must allow for the correction of one bit-error, find the least possible length of the codewords. 8.4. Suppose it is required to construct a linear code with n = 12 and δ = 3. Find an upper bound for the information rate of such a code. 8.5. Given words x, y ∈ F2 n let x ∗ y denote the word that has 1 in any position if and only if both x and y have 1 in that position. Prove that w(x + y) = w(x) + w(y) − 2w(x ∗ y). Deduce that the set of all words with even weight in F2 n is a linear code.

8.6. Let B(n, δ) denote the maximum dimension of a linear code in F2 n with minimum distance δ. Use the packing bound to show that B(6, 3) ≤ 3,

B(7, 3) ≤ 4,

B(8, 3) ≤ 4.

8.2 Construction of linear codes using matrices In Chapter 7 we studied the problem of transmitting information at a given rate, while reducing the probability of a mistake. The key idea was to split the original stream of bits into blocks of size k and assign a codeword of length n to each of the 2k possible blocks. In other words, we require an encoding function F2 k → F2 n . Given that F2 k and F2 n are vector spaces, an obvious candidate for such a function is a linear transformation, defined by a matrix. For the purposes of matrix algebra it is often convenient to replace a word x, regarded as a row vector, by the corresponding column vector x′ (the transpose of x). Thus, if E is a n × k matrix over F2 and y is in F2 k then then the word x defined by x′ = Ey ′ is in F2 n . The matrix E defines an encoding function F2 k → F2 n .

Example 8.6 In Example 7.5 we gave a rule that assigns to each block y1 y2 y3 of size 3 a codeword x1 x2 x3 x4 x5 x6 of length 6. Express this rule in matrix form.

8.2 Construction of linear codes using matrices

127

Solution The rule for x4 (for example) is that x4 = 0 if y1 = y2 and x4 = 1 otherwise. Using the algebraic structure of the field F2 this can be expressed by the linear equation x4 = y1 + y2 . In the same way, we have the equations x5 = y2 + y3 , x6 = y1 + y3 . All the equations can be written as a single matrix equation: ⎞ x1 ⎜ x2 ⎟ ⎜ ⎟ ⎜ x3 ⎟ ⎜ ⎟ ⎜ x4 ⎟ ⎜ ⎟ ⎝x ⎠ ⎛

5

x6

⎞ y1 ⎜ y2 ⎟ ⎟ ⎜ ⎜ y3 ⎟ ⎟ ⎜ ⎜ y1 + y2 ⎟ ⎟ ⎜ ⎝y +y ⎠ ⎛

=

2

=

3

⎞ y1 E ⎝ y2 ⎠ , y3 ⎛

y1 + y3

where E is the 6 × 3 matrix given by ⎛ 1 ⎜0 ⎜ ⎜0 E=⎜ ⎜1 ⎜ ⎝0 1

0 1 0 1 1 0

⎞ 0 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 1⎠ 1

As in Example 7.5, we can check that the resulting code C ⊆ F2 6 has parameters n = 6, k = 3, and δ = 3, and hence it is a 1-error-correcting code with rate 1/2. In general, given any n × k matrix E over F2 , let C be the set {x ∈ F2 n | x′ = Ey ′ for some y ∈ F2 k }. If x and w are in C, say x′ = Ey ′ and w′ = Ez ′ , then (x + w)′ = E(y + z)′ , so x + w is also in C. Thus C is a linear code. Technically, it is the image of E. Essentially, we now have the answer to the practical problem of encoding a stream of bits. The stream is split into blocks y of length k, and the codeword for y is x, where x′ = Ey ′ , for a suitable matrix E. Of course, we must try to ensure that resulting code has good error-correction properties, and that there is a workable method of implementing the MD rule. For these purposes a slight modification of the construction is useful, as we shall see in the next section.

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8. Linear codes

EXERCISES 8.7. Suppose we encode a block of bits y1 y2 . . . yk by setting xi = yi for i = 1, 2, . . . k, and defining the parity check bit xk+1 as follows:  0 if an even number of the yi ’s are 1; xk+1 = 1 if an odd number of the yi ’s are 1. Write down the matrix E such that x′ = Ey ′ . 8.8. Show that in the code defined in previous exercise every word has even weight. 8.9. Find the information rate ρ and the minimum distance δ for the parity check code described in Exercise 8.7.

8.3 The check matrix of a linear code In the previous section we described a method of coding a stream of bits by dividing it into blocks of length k and applying an n × k matrix E to each block y. In Example 8.6 the transformation x′ = Ey ′ was defined by the linear equations x1 = y1 , x2 = y2 , x3 = y3 , x4 = y1 + y2 ,

x5 = y2 + y3 ,

x6 = y1 + y3 .

The variables y1 , y2 , y3 can be eliminated from the equations, giving x1 + x2 + x4 = 0,

x2 + x3 + x5 ,

x1 + x3 + x6 = 0.

(Remember that −1 = 1 in the field F2 .) These equations can be written in the form of a matrix equation Hx′ = 0′ , where ⎛ ⎞ 1 1 0 1 0 0 H = ⎝0 1 1 0 1 0⎠. 1 0 1 0 0 1 Hence we have an alternative means of defining the code, as the set of x such that Hx′ = 0′ . In Linear Algebra this set is known as the kernel or null space of H; it is clearly a subspace of F2 n , so we have a linear code. The general situation is as follows.

8.3 The check matrix of a linear code

129

Lemma 8.7 Let E be an n × k matrix over F2 , of the form   I , A where I is the identity matrix with size k, and A is any (n − k) × k matrix. Then the code {x ∈ F2 n | x′ = Ey ′ for some y ∈ F2 k } can also be defined as {x ∈ F2 n | Hx′ = 0′ }, where H is the (n − k) × n matrix (A I). (Here I is the identity matrix with size n − k.)

Proof When E has the stated form the condition x′ = Ey ′ is  ′   ′  y y ′ x′ = = ( A I ) , so Hx = Ay ′ + Ay ′ . Ay ′ Ay ′ This reduces to 0′ because 0 + 0 and 1 + 1 are both 0 in F2 . Conversely, when H has the stated form and Hx′ = 0′ , let x = (a b) so that  ′ a ′ ′ 0 = Hx = ( A I ) = Aa′ + b′ . b′ Thus Aa′ = b′ , and x is in the image of E since    ′  I a ′ ′ Ea = = x′ . a = A Aa′

Definition 8.8 (Check matrix) A matrix H over F2 with m rows and n columns is the check matrix for the linear code C = {x ∈ F2 n | Hx′ = 0′ }.

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8. Linear codes

In the definition H can be any m × n matrix over F2 . Suppose, as in Lemma 8.7, that H has the standard form (A I), where A has n − m columns and I has m columns. Then the corresponding code has dimension k = n − m. To prove this, let ⎛ ⎞ a11 a12 · · · a1k 1 0 ··· 0 ⎜ a21 a22 · · · a2k 0 1 ··· 0⎟ ⎜ ⎟ ⎜ H = (A I) = ⎜ . . ··· . . . ··· . ⎟ ⎟. ⎝ . . ··· . . . ··· . ⎠ am1 am2 · · · amk 0 0 ··· 1 Then the equations Hx′ = 0′ are a11 x1 + a12 x2 + · · · + a1k xk + xk+1 = 0 a21 x1 + a22 x2 + · · · + a2k xk + xk+2 = 0 ...

...

am1 x1 + am2 x2 + · · · + amk xk + xn = 0. These equations can be rearranged so that the values xk+1 , xk+2 , . . . xn are defined in terms of the values x1 , x2 , . . . , xk : xk+1 = a11 x1 + a12 x2 + · · · + a1k xk xk+2 = a21 x1 + a22 x2 + · · · + a2k xk ...

...

xn = am1 x1 + am2 x2 + · · · + amk xk . In a typical codeword x = x1 x2 . . . xn , we say that x1 , x2 , . . . , xk are message bits xk+1 , xk+2 , . . . , xn are check bits. If the message bits are given arbitrary values, the values of the check bits are determined by the equations displayed above. Since there are 2k possible values of the message bits, the dimension of the code is k.

Example 8.9 Make a list of the codewords defined by the check matrix   1 0 1 0 H= . 1 1 0 1 What are the parameters of this code?

8.4 Constructing 1-error-correcting codes

Solution

131

If x = x1 x2 x3 x4 then the condition Hx′ = 0′ means that x1 + x3 = 0,

x1 + x2 + x4 = 0.

We can rewrite these equations so that x3 and x4 are given in terms of x1 and x2 : x4 = x1 + x2 . x3 = x1 , The codewords can be found by giving all possible values to x1 , x2 and using the equations to calculate x3 , x4 . x1 0 0 1 1

x2 0 1 0 1

x3 0 0 1 1

x4 0 1 1 0

weight 0 2 3 3

The code has parameters n = 4, k = 2, and the minimum distance is equal to the minimum nonzero weight, which is δ = 2.

EXERCISES 8.10. Make a list of the codewords defined ⎛ 1 0 1 H = ⎝1 1 0 0 1 0

by the check matrix ⎞ 0 0 1 0⎠. 0 1

What are the parameters of this code? 8.11. Write down a check matrix for the parity check code described in Exercise 8.7. 8.12. The code C1 = {00, 01, 10, 11} may be used to represent four messages {a, b, c, d}. The code Cr is obtained by repeating each word in C1 r times: for example, in C3 the message b is represented by 010101. Show that Cr is a linear code, find its parameters (n, k, δ) and write down a check matrix for it.

8.4 Constructing 1-error-correcting codes The following theorem describes a very simple way of constructing a check matrix for a 1-error-correcting code.

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8. Linear codes

Theorem 8.10 The code C defined by a check matrix H is a 1-error-correcting code provided that •1 no column of H consists entirely of 0’s; and •2

no two columns of H are the same.

Proof Suppose there is a codeword c ∈ C with weight 1, say c = 0 · · · 010 · · · 0, where the ith bit is 1. Then, according to the rule for matrix multiplication (Figure 8.1), Hc′ is equal to the ith column of H.

··· ∗ ⎜··· ∗ ⎜ Hc′ = ⎜ ⎜ ⎝··· ∗ ··· ∗ ⎛

⎛ ⎞ 0 ⎛ ⎞ ∗ ··· ⎜ . ⎟ ⎜ ⎟ ⎜∗⎟ ⎜0⎟ ···⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎟⎜1⎟ = ⎜ ⎟. ⎜ ⎟ ⎟⎜ ⎟ ⎟ ⎝∗⎠ ···⎠⎜ ⎜0⎟ ⎠ ⎝ . ∗ ··· 0 ⎞

Figure 8.1 Calculation of Hc′ when c = 0 · · · 010 · · · 0 But if c is a codeword Hc′ = 0′ , so the ith column must be the zero column 0′ , contradicting condition •1. Suppose there is a codeword of weight 2, say c = 0 · · · 010 · · · 010 · · · 0, where the ith and jth bits are 1. Then, by a similar calculation (Figure 8.2), Hc′ is equal to the sum of the ith and jth columns of H. But if c is a codeword, then Hc′ = 0′ . Since we are using arithmetic in the field F2 , this means that the ith column and the jth column are equal, contradicting condition •2. Hence, if the conditions are satisfied, any nonzero codeword must have weight at least 3, and we have a 1-error correcting code.

8.4 Constructing 1-error-correcting codes

··· ∗ ··· ⎜··· ∗ ··· ⎜ Hc′ = ⎜ ⎜ ⎝··· ∗ ··· ··· ∗ ··· ⎛

† † † †

133

⎛ ⎞ 0 ⎜.⎟ ⎜ ⎟ ⎜0⎟ ⎟ ⎞⎜ ⎛ ⎞ ⎛ ⎞ ⎟ ··· ⎜ ∗ † ⎜1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ···⎟ 0 ∗ ⎟ ⎟⎜ ⎜ ⎟ ⎜†⎟ ⎟ ⎟⎜ ⎜ ⎟ ⎜ .⎟ = ⎜ ⎟+⎜ ⎟ ⎟⎜ ⎟. ⎜ ⎟ ⎠ ⎝∗⎠ ⎝†⎠ ··· ⎜0⎟ ⎜ ⎟ ⎟ ··· ⎜ ∗ † ⎜1⎟ ⎜0⎟ ⎜ ⎟ ⎝.⎠ 0

Figure 8.2 Calculation of Hc′ when c = 00 · · · 010 · · · 010 · · · 0.

Example 8.11 In Example 7.4 we showed that the smallest possible values of n and k for a 1-error-correcting code with information rate 0.8 are n = 25, k = 20. How can such a code be constructed using a check matrix? Solution We require H to be an m × n matrix with m = n − k = 5 rows. According to Theorem 8.10, the n = 25 columns must be distinct, and not the zero column. In fact there are 25 − 1 = 31 possible columns and any 25 of them can be chosen. If H is in standard form the last five columns will be the five words of weight 1, and the first 20 columns can be any other nonzero words. The matrix algebra illustrated in Figure 8.1 provides a remarkably simple way of implementing the MD rule in some circumstances.

Theorem 8.12 Let C ⊆ F2 n be a linear code defined by a check matrix H. Suppose that one bit-error is made in transmitting a codeword, the received word being z. Then the error has occurred in the ith bit of z, where i is determined by the fact that Hz ′ is equal to the ith column of H.

Proof Suppose that the codeword sent is c and the error is made in the ith bit. Then the received word is z = c + w, where w is the word 0 · · · 010 · · · 0 with 1 in the ith place. Since c is a codeword, Hc′ = 0′ , and (as in Figure 8.1) Hw′ is the

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8. Linear codes

ith column of H. Hence Hz ′ = H(c + w)′ = Hc′ + Hw′ is the ith column of H. Assuming that not more than one bit-error is made in transmitting each codeword, the procedure shown in Figure 8.3 can be used. ? Hz ′ = 0′ yes

z is a codeword

no ? Hz ′ = column i yes

correct ith bit

no more than 1 bit error made

Figure 8.3 Processing a received word z for a code with check matrix H

For each received word z the Receiver should calculate Hz ′ . If Hz ′ = 0, z is itself the intended codeword. If Hz ′ is equal to a column of H, say the ith column, then the intended codeword is obtained by changing the ith bit in z. In any other case, at least two bit-errors have occurred.

EXERCISES 8.13. Construct a 1-error-correcting code C ∈ F2 6 with |C| = 8. 8.14. Find the smallest values of n and k for which a 1-error correcting code with information rate 0.75 can exist, and explain how to define such a code by means of a check matrix. 8.15. Write down all the codewords belonging to the linear code with the following check matrix, and find the parameters (n, k, δ) for this code.

8.5 The decoding problem

135

1 ⎜0 ⎜ ⎝1 0 ⎛

1 0 0 0

0 0 1 0

1 1 1 0

0 1 0 0

0 0 0 1

⎞ 1 1⎟ ⎟. 1⎠ 1

8.16. In Exercise 8.2 we showed that it is impossible to encode 128 different messages in such a way that one error is corrected, using binary words of length 10. Explain how to construct a linear code satisfying the conditions, using words of length 11. 8.17. A codeword from the code defined by the following check matrix is sent, and the word 111010 is received. ⎞ ⎛ 1 1 1 1 0 0 ⎝1 1 0 0 1 0⎠. 1 0 1 0 0 1 What was the intended codeword, assuming that only one error has been made? 8.18. In the preceding exercise, how many received words cannot be corrected on the assumption that at most one error has been made?

8.5 The decoding problem In general, there is no easy way of implementing the MD decision rule. Given a set of codewords C ⊆ Bn and a word z ∈ Bn the problem is to find a codeword c ∈ C that is nearest to z, in the sense of Hamming distance. Of course, it can be done by the brute-force method of running through a list of all the codewords, but that is not an efficient method. When C is defined by means of an algebraic construction, there may be ways of improving on the brute-force approach. For example, if the code is defined by a check matrix and we can assume that only one bit-error has occurred, the simple rule stated in Theorem 8.12 can be used. More generally, the decoding problem can often be simplified by using a a technique known as syndrome decoding.

Definition 8.13 (Syndrome) Let C be the linear code defined by a check matrix H. For any word z ∈ F2 n the syndrome of z is s, where Hz ′ = s′ .

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8. Linear codes

If z is a codeword then the syndrome of z is the zero word. If z is the result of making a single error, in the ith bit of a codeword, then the syndrome is the word in the ith column of H.

Lemma 8.14 Two words y, z ∈ F2 n have the same syndrome with respect to C if and only if y = z + c, for some c ∈ C.

Proof It follows from the definitions that Hy ′ = Hz ′

⇐⇒

H(y ′ − z ′ ) = 0′

⇐⇒

y−z ∈C

⇐⇒

y = z + c.

The set of y ∈ F2 n such that y = z + c for some c ∈ C is known as the coset of z with respect to C, and is denoted by z + C. We recall from elementary Group Theory that two cosets are either disjoint or identical (they cannot partly overlap), so the distinct cosets form a partition of F2 n .

Example 8.15 List the cosets of the code C ⊆ F2 4 defined by the check matrix   1 0 1 0 H= . 1 1 0 1 Solution In Example 8.9 we showed that C contains four codewords, 0000, 0101, 1011, 1110. Since |C| = 4 and |F2 4 | = 24 = 16, there are 16/4 = 4 distinct cosets. The coset 0000 + C is C itself. A distinct coset z + C can be constructed by choosing any z that is not in this coset, such as z = 1000. Continuing in this way, we obtain the four distinct cosets, as listed below. 0000 + C 0000 0101 1011 1110

1000 + C 1000 1101 0011 0110

0100 + C 0100 0001 1111 1010

0010 + C 0010 0111 1001 1100

8.5 The decoding problem

137

If the code has word-length n and dimension k, the number of cosets is 2n /2k = 2n−k = 2m , where m is the number of rows of H. Since each syndrome is a word of length m, and different cosets have different syndromes, all possible words in F2 m occur as syndromes. It is convenient to arrange them in a definite order, such as the dictionary order (Definition 4.15).

Definition 8.16 (Coset leader, syndrome look-up table) A coset leader is a word of least weight in its coset; if there are several possibilities, we choose one for definiteness. A syndrome look-up table is an ordered list of pairs (s, f ) such that, for each syndrome s, f is the coset leader for the coset that has syndrome s. In the list of cosets given in Example 8.15, the first element of each coset has least weight, and can be chosen as the coset leader. However, in the third coset there are two words of weight 1, and either of them could be chosen. If we choose 0100 from this coset, and arrange the syndromes in dictionary order, the syndrome look-up table is as follows. syndrome : coset leader :

00 01 10 11 0000 0100 0010 1000

The syndrome decoding method is based on the assumption that the Receiver knows the check matrix H for a code C and has a copy of the look-up table. In that case, the following decision rule σ : F2 n → C can be applied. • For a given received word z calculate the syndrome s, using s′ = Hz ′ .

• Look up the coset leader f corresponding to s. • Define σ(z) = z + f .

Theorem 8.17 With the notation used above, (i) σ(z) = z + f is a codeword; (ii) there is no codeword c ∈ C such that d(z, c) < d(z, z + f ).

Proof (i) Since f and z have the same syndrome s, they are in the same coset, and f = z + c∗ for some c∗ ∈ C. Thus z + f = z + (z + c∗ ) = (z + z) + c∗ = c∗ ,

which is in C.

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8. Linear codes

(ii) We have d(z, z + f ) = w(z + z + f ) = w(f ). For any codeword c, d(z, c) = w(z + c), so if d(z, c) < d(z, z + f ) it must follow that w(z + c) < w(f ). But z + c and z are in the same coset, so z + c and f are in the same coset, and since f is a coset leader, w(z + c) ≥ w(f ). The first part of the theorem shows that σ is a valid decision rule. The second part shows that it is an MD rule. The choice of coset leader corresponds to choosing one of the codewords nearest to z as the codeword σ(z).

Example 8.18 The check matrix ⎛

1 0 H = ⎝1 1 0 1

1 0 0 1 0 0

⎞ 0 0⎠ 1

defines the code C = {00000, 01011, 10110, 11101}. Construct a syndrome lookup table for C and use it to determine σ(10111). Solution The cosets can be constructed in the usual way. The first coset is 00000 + C = C. This does not contain 10000 (for example), so the next coset is 10000 + C. Continuing in this way we obtain eight cosets 00000 + C

10000 + C

01000 + C

00100 + C

00010 + C

00001 + C

00101 + C

00111 + C.

Note that the representatives used to construct the cosets are not necessarily the coset leaders. To find the coset leaders we must choose one word from each coset that has minimum weight. For example, the coset 00111 + C contains the words 00111, 01100 10001, and 11010. Thus the coset leader could be either 01100 or 10001. Suppose we choose 01100 as the coset leader for this coset, and make similar choices for the other cosets. For each coset leader f , the syndrome s for all words in that coset is given by s′ = Hf ′ . The following is a syndrome look-up table for C (other choices for the coset leaders are possible). 000 001 010 011 100 101 110 111 00000 00001 00010 01000 00100 00101 10000 01100 Suppose z = 10111 is the received word. Since Hz ′ = 001′ the corresponding coset leader is f = 00001 and the Receiver will decide that σ(z) = z + f = 10110.

8.5 The decoding problem

139

It should not be thought that syndrome decoding, in itself, is always an improvement on the brute-force method of implementing the MD rule. If the received word z is compared with each codeword in order to find the nearest one, then, for a linear code of dimension k, 2k comparisons are needed. With syndrome decoding the Receiver must have a copy of the look-up table, which contains 2n−k entries, and this too may be a large number, so the method cannot be regarded as ‘efficient’ in a strict sense. Fortunately, when specific codes are constructed by algebraic methods (as in the next chapter) there may be better ways of implementing the rule.

EXERCISES 8.19. Construct a syndrome look-up table for the code defined by the check matrix ⎛

1 ⎝0 1

0 1 1 0 1 0

⎞ 0 0 1 0⎠. 0 1

Use your table to determine the codewords corresponding to the following received words: 11111,

11010,

01101,

01110.

8.20. Suppose the Sender uses the code C = {00000, 01011, 10110, 11101} and the Receiver uses the syndrome look-up table for C given in Example 8.18. Given that the probability of a bit-error in transmission is e, estimate M0 , the probability of a mistake when the codeword 00000 is sent. 8.21. Denote by hi the column 4-vector corresponding to the binary representation of the integer i (1 ≤ i ≤ 15). For example, h9 = [1001]′ . Let H be the 8 × 15 binary matrix with the first four rows h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 , and the last four rows h5 h3 h1 h1 h8 h15 h1 h15 h8 h5 h5 h8 h3 h15 h3 .

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8. Linear codes

H is the check matrix for a code C ⊆ F2 15 . A codeword c ∈ C is transmitted and the received word is z = 100 100 100 100 000. Calculate the syndrome of z. How do we know that z is not a codeword, and that more than one bit-error has been made? Assuming that exactly two bit-errors have been made, which two bits should be corrected? [The general construction of which this is an example will be discussed in the next chapter.]

Further reading for Chapter 8 The standard texts by MacWilliams and Sloane [6.2], and Pless and Huffman [6.3] are recommended for background reading on error-correcting codes.

9 Algebraic coding theory

9.1 Hamming codes In the previous chapter we showed that algebraic methods can be used to construct useful codes. To set the scene for further developments we begin by describing an important family of codes, discovered by R.W. Hamming in 1950. According to Theorem 8.10 a binary linear code with δ ≥ 3 can be defined by a check matrix in which all the columns are distinct and non-zero. If the number m of rows is given, then there are 2m column vectors of length m, and so the maximum number of distinct non-zero columns is 2m − 1.

Definition 9.1 (Hamming code) A code defined by a binary check matrix with m rows and 2m − 1 distinct non-zero columns is a binary Hamming code. Note that the order in which the columns are listed is unimportant, although in practice a definite order is obviously needed. Rearranging the rows just involves a corresponding rearrangement of the bits in each codeword, and does not affect the parameters of the code. Two codes that are related in this way are said to be equivalent.

N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 9, 

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Example 9.2 Find the check matrix and the parameters of the Hamming code with m = 3. Solution There are 23 − 1 = 7 columns. One way of listing them would be to put them in the dictionary order (Definition 4.15). But, as we shall see shortly, there are good reasons for using the numerical order, which is the order corresponding to the binary representations of the numbers 1, 2, 3, 4, 5, 6, 7: ⎛ ⎞ 0 0 0 1 1 1 1 H1 = ⎝ 0 1 1 0 0 1 1 ⎠ . 1 0 1 0 1 0 1 Alternatively, we could arrange the columns so that those with weight 1 come last, obtaining a check matrix in the standard form: ⎛ ⎞ 0 1 1 1 1 0 0 H2 = ⎝ 1 0 1 1 0 1 0 ⎠ . 1 1 0 1 0 0 1 Clearly, the codes defined by the matrices H1 and H2 are equivalent. The parameters (n, k, δ) are easily found. The word-length is n = 7, and the dimension is k = n − m = 4. The minimum distance δ is at least 3, since all the columns of H1 are distinct and non-zero, and it is easy to check that 1110000 is a codeword with weight 3, so δ is exactly 3. Generally, for each m ≥ 3, there is a binary Hamming code with parameters n = 2m − 1,

k = 2m − 1 − m,

δ = 3.

These codes have a very special property.

Theorem 9.3 In a binary Hamming code, every word in F2 n is either a codeword or is at distance 1 from exactly one codeword.

Proof There are 2k codewords, and for each c ∈ C the size of the neighbourhood N1 (c) is 1 + n. We know that δ = 3, so these neighbourhoods are disjoint. Substituting the appropriate values of the parameters, we have m

2k (1 + n) = 2k × 2m = 2m+k = 22

−1

= 2n .

This means that the 2k codewords c ∈ C determine disjoint neighbourhoods N1 (c) that completely cover F2 n , as claimed.

9.1 Hamming codes

143

Generally, an r-error correcting code has the property that δ ≥ 2r + 1, so the neighbourhoods Nr (c) (c ∈ C) are mutually disjoint. However, there will usually be ‘gaps’ not covered by these neighbourhoods, so that some words are not in any of the neighbourhoods Nr (c).

Definition 9.4 (Perfect code) An r-error-correcting code C is said to be a perfect code if every word is in one of the neighbourhoods Nr (c) for some c ∈ C. Theorem 9.3 asserts that a binary Hamming code is a perfect code with r = 1. In the case m = 3 there are 24 = 16 codewords c, and each neighbourhood N1 (c) contains 1 + 7 = 8 words. These 16 neighbourhoods are disjoint and, since 16 × 8 = 128 is the total number of words, the code is perfect (see Figure 9.1).

Figure 9.1 The Hamming code with parameters (7, 4, 3) is perfect

The main reason for using algebraic methods is that the algebraic structure can provide good algorithms for encoding and decoding. For the binary Hamming codes, we can use the following techniques. •Encoding Suppose the columns of the check matrix H are arranged in numerical order. Then the bits x1 , x2 , x4 , . . . , x2m−1 are check bits, and we can write down the equations that define them in terms of the remaining (message)

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9. Algebraic coding theory

bits. For example, in the case m = 3 the equations derived from H1 are x1 = x3 + x5 + x7 ,

x2 = x3 + x6 + x7 ,

x4 = x5 + x6 + x7 .

Sometimes it is convenient to use the first k bits as the message bits and the remaining n − k bits as the check bits. This corresponds to writing the check matrix in the standard form H2 , whose columns are obtained from those of H1 by the permutation 1 → 7, 2 → 6, 3 → 1, 4 → 5, 5 → 2, 6 → 3, 7 → 4. Applying this permutation, and reordering the equations, the check bits are given by x5 = x2 + x3 + x4 ,

x6 = x1 + x3 + x4 ,

x7 = x1 + x2 + x4 .

•Decoding When the columns of the check matrix are arranged in numerical order, the syndrome decoding rule takes a very simple form. First, we observe (Exercise 9.3) that there are n + 1 cosets 0 + C,

e1 + C,

e2 + C,

...,

en + C,

where ei denotes the word 0 · · · 010 · · · 0 in which the ith bit is 1. The syndrome of the coset ei + C is He′i , which is equal to the ith column of H. But this is just the binary representation of i. So the rule for correcting a single error is: if the syndrome of the received word z is the binary representation of i, then the ith bit is in error.

EXERCISES 9.1. Write down the check matrix for the Hamming code of length 15, using the numerical order of the columns. How many codewords are there? Which of the following words are codewords? 011010110110000,

100000100000011,

110010110111111.

Correct those that are not codewords, assuming that only one error has been made. 9.2. Show that if the check matrix for a Hamming code is arranged so that the columns are in numerical order, then the word 11100 · · · 00 is a codeword. 9.3. Prove that the number of cosets of the the binary Hamming code C with words of length n is n + 1, and show that a complete set of coset representatives is the set of n + 1 words 0, e1 , e2 , . . . en , as defined above.

9.2 Cyclic codes

145

9.2 Cyclic codes The Hamming codes provide a family of examples in which the dimension k of the of the code, and thus its size |C| = 2k , can be as large as we please, while the minimum distance is constant. Question: Is it possible to construct explicitly families of linear codes in which both the dimension and the minimum distance can be as large as we please? In order to answer this question positively we shall give (i) a general method for constructing linear codes, based on simple algebraic ideas, and (ii) a specific construction using the general method. In the rest of this chapter we shall use the notation a = a0 a1 . . . an−1 to represent a typical word in F2 n . The cyclic shift of a is the word ˆ = an−1 a0 a1 . . . an−2 . a

Definition 9.5 (Cyclic code) A set C ⊆ F2 n is said to be a cyclic code if (i) it is a linear code, and

(ii) ˆ c is in C whenever c is in C, that is, c0 c1 c2 . . . cn−1 ∈ C

=⇒

cn−1 c0 c1 . . . cn−2 ∈ C.

The definition implies that if C is cyclic and c ∈ C then the words ci ci+1 · · · cn−1 c0 · · · ci−1 obtained from c by any number of cyclic shifts are also in C.

Example 9.6 Which of the following codes are cyclic? C1 = {000, 100, 010, 001},

C2 = {0000, 1010, 0101, 1111}.

Solution A cyclic code must be linear, and it must be closed under the cyclic shift operation. Thus C1 is not a cyclic code, because it is not a linear code (100 + 010 = 110, which is not C1 , for example). On the other hand, C2 is a cyclic code. It is easy to check both conditions: for example, the cyclic shift of the codeword 1010 is the codeword 0101.

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9. Algebraic coding theory

There is a simple way of representing a cyclic shift in algebraic terms. An expression a(x) = a0 + a1 x + a2 x2 + · · · + ad xd in which a0 , a1 , a2 , . . . , ad are in F2 , is said to be a polynomial with coefficients in F2 . If ad = 0 then the degree of a(x) is d. Polynomials can be added and multiplied according to the rules we learn in elementary algebra, and with these rules they form a ring, denoted by F2 [x]. Clearly, there is a correspondence between the polynomial a0 + a1 x + · · · + an−1 xn−1 in F2 [x], and the word a0 a1 . . . an−1 in F2 n .

ˆ of a is represented by the polynoIn this correspondence, the cyclic shift a mial a ˆ(x), where a ˆ(x)

= an−1 + a0 x + · · · + an−2 xn−1 = x(a0 + a1 x + · · · + an−1 xn−1 ) − an−1 (xn − 1) = xa(x) − an−1 (xn − 1).

We shall retain the minus signs for the sake of clarity but, since the coefficients belong to F2 we could rewrite all the minus signs as plus signs. The result of the calculation is that a ˆ(x) is equal to xa(x), except for a multiple of xn − 1; in other words a ˆ(x) is equal to xa(x) modulo xn − 1. Suppose we define a new rule for multiplying polynomials, as follows. Given a(x) and b(x), multiply them in the usual way, and then reduce modulo xn − 1. This means that we take xn − 1 to be the same as 0, and so the reduction is equivalent to replacing xn by 1, xn+1 by x, xn+2 by x2 , and so on. With this rule it is clear that any polynomial a(x) in F2 [x] can be reduced to a polynomial with degree less than n. We shall denote by V n [x] the ring of polynomials with coefficients in F2 , using multiplication modulo xn − 1. Technically, V n [x] is a quotient ring of F2 [x]. We have constructed the ring V n [x] so that it is in bijective correspondence with F2 n , specifically a(x) ↔ a. Furthermore, if a(x) and b(x) correspond to a ˆ, the and b, then a(x) + b(x) corresponds to a + b and xa(x) corresponds to a first cyclic shift of a.

Example 9.7 Write down the polynomials in V 6 [x] that correspond to the words 110101 and 010110, and find their product as elements of V 6 [x].

9.2 Cyclic codes

147

Solution 110101 is represented by 1 + x + x3 + x5 and 010110 is represented by x + x3 + x4 . Multiplying and putting x6 = 1, x7 = x, and so on, we obtain (1 + x + x3 + x5 )(x + x3 + x4 ) = (x + x3 + x4 ) + (x2 + x4 + x5 ) + (x4 + x6 + x7 ) + (x6 + x8 + x9 ) = x + x2 + x3 + x4 + x5 + x7 + x8 + x9 = x + x2 + x3 + x4 + x5 + 1 + x + x2 = 1 + x3 + x4 + x5 . We shall now prove that a cyclic code in F2 n corresponds to a particular kind of subset of V n [x]. Let R be a ring with commutative multiplication. A subset S of R is said to be an ideal if (i) a, b ∈ S ⇒ a + b ∈ S (ii) r ∈ R and a ∈ S ⇒ ra ∈ S.

In other words, an ideal S is closed under addition, and under multiplication by any element of R.

Theorem 9.8 A binary code with codewords of length n is cyclic if and only if it corresponds to an ideal in V n [x].

Proof Suppose C is a cyclic code, represented as a subset of V n [x]. Since C is linear, if a(x) and b(x) are in C so is a(x) + b(x), and the first condition for an ideal is satisfied. Since xa(x) represents the first cyclic shift of a(x), it follows that xa(x) is in C whenever a(x) is in C. By repeating the same argument xi a(x) is in C whenever a(x) is in C, for any i ≥ 0. Any polynomial p(x) is the sum of a number of powers xi and so, since C is linear, p(x)a(x) is in C. Hence C is an ideal. Conversely, if C is an ideal, condition (i) tells us that it represents a linear code. Condition (ii) tells us (in particular) that xa(x) is in C whenever a(x) is, so C is cyclic. It follows from the theorem that the construction of cyclic codes of length n is equivalent to the construction of ideals in V n [x]. There is a very simple way of constructing such ideals. Let f (x) be any polynomial in V n [x]. The set of all multiples of f (x) in n V [x] is clearly an ideal, for if a(x) and b(x) are multiples of f (x), so are

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9. Algebraic coding theory

a(x) + b(x) and p(x)a(x) for any p(x). We denote this ideal by f (x) and refer to it as the ideal generated by f (x).

Example 9.9 Construct the ideal generated by f (x) = 1 + x2 in V 3 [x], and write down the corresponding code C ⊆ F2 3 .

Solution Multiplying f (x) in turn by each element p(x) of V 3 [x] and reducing modulo x3 − 1, we obtain the following table. p(x) 0 1 x 1+x x2 1 + x2 x + x2 1 + x + x2

p(x)f (x) mod(x3 − 1) 0 1 + x2 1+x x + x2 x + x2 1+x 1 + x2 0

So the ideal 1 + x2  has just four elements 0,

1 + x,

x + x2 ,

1 + x2 .

The corresponding code in F2 3 is C = {000, 110, 011, 101}.

EXERCISES 9.4. Which of the following codes are cyclic? C1 = {0000, 1100, 0110, 0011, 1001}, C2 = {0000, 1100, 0110, 0011, 1001, 1010, 0101, 1111}. 9.5. Write down the codewords of the cyclic code corresponding to the ideal 1 + x + x2  in V 3 [x], and find a check matrix for this code. 9.6. Show that the ideal 1 + x in V 5 [x] corresponds to the code in F2 5 containing all the words of even weight.

9.7. Does the result in the previous exercise hold when 5 is replaced by an arbitrary integer n ≥ 2?

9.3 Classification and properties of cyclic codes

149

9.3 Classification and properties of cyclic codes The next theorem shows that every cyclic code corresponds to an ideal generated by a polynomial.

Theorem 9.10 Let C = {0} be a cyclic code (ideal) in V n [x]. Then there is a polynomial g(x) in C such that C = g(x).

Proof Since C is not {0} it contains a non-zero polynomial g(x) of least degree. If f (x) is any element of C let q(x) and r(x) be the quotient and remainder when f (x) is divided by g(x): f (x) = g(x)q(x) + r(x), where deg r(x) < deg g(x). Because g(x) is in the ideal C it follows that g(x)q(x) is in C, and since f (x) is also in C, g(x)q(x) − f (x) = r(x) is in C. This contradicts the definition of g(x) as a non-zero polynomial of least degree in C, unless r(x) = 0. Hence f (x) = g(x)q(x), which means that C = g(x), as claimed. The polynomial g(x) obtained in the proof is uniquely determined by the property that it has the least degree in C. For if g1 (x) and g2 (x) both have this property, they have the same degree and their leading coefficients are both 1. (Since the coefficients are in F2 the only possible coefficients are 0 and 1.) Furthermore, since C is an ideal g1 (x) − g2 (x) is also in C, and its degree is less than the degree of g1 (x) and g2 (x). This is a contradiction unless g1 (x) − g2 (x) = 0. In Example 9.9 we listed the ideal in V 3 [x] generated by 1 + x2 . Inspection of the list shows that in this case the unique non-zero polynomial of least degree is 1 + x, and Theorem 9.10 tells us that this polynomial is also a generator for the ideal. In general, a cyclic code C will have many generators, but only one of them will have the least degree in C. We shall refer to the unique polynomial with this property as the canonical generator of C.

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9. Algebraic coding theory

Theorem 9.11 The canonical generator g(x) of a cyclic code C in V n [x] is a divisor of xn − 1 in F2 [x].

Proof Dividing xn − 1 by g(x) in F2 [x], we obtain a quotient and remainder: xn − 1 = g(x)h(x) + s(x),

where deg s(x) < deg g(x). This equation implies that, in the quotient ring V n [x], s(x) = g(x)h(x). Since C is the ideal in V n [x] generated by g(x) it follows that s(x) is in C. This contradicts the fact that g(x) has the least degree in C, unless s(x) = 0. Hence xn − 1 = g(x)h(x) in F2 [x], as claimed. We shall now explain how the canonical generator g(x) determines the dimension of a cyclic code, and a check matrix for it. Since g(x) is a divisor of xn − 1, we have g(x)h(x) = xn − 1 in F2 [x]. Let g(x) = g0 + g1 x + · · · + gn−k xn−k ,

h(x) = h0 + h1 x + · · · + hk xk ,

where the coefficients g0 , h0 , gn−k , hk must all be 1, since the product of the polynomials is xn − 1. Let g be the word g = g0 g1 . . . gn−k 0 0 . . . 0, in F2 n , and let h∗ be the word whose first k + 1 bits are the coefficients of h(x) in reverse order, followed by n − k − 1 zeros: h∗ = hk hk−1 · · · h0 0 0 · · · 0.

Let H be the (n − k) × n matrix whose rows are h∗ and the first n − k − 1 cyclic shifts of h∗ : ⎛ ⎞ hk hk−1 h0 0 0 0 ⎜ 0 hk . h1 h0 0 0 ⎟ ⎜ ⎟ ⎜ 0 h1 h0 0 ⎟ 0 . . h2 ⎜ ⎟. H =⎜ ⎟ . . . ⎜ ⎟ ⎝ ⎠ . . . h0 0 0 hk hk−1

Lemma 9.12 Let g(i) be the row-vector 0 0 · · · 0g0 g1 · · · gn−k 0 0 · · · 0 where there are i zeros at the beginning and k − 1 − i zeros at the end, that is, the cyclic shift of g ′ = 0′ (0 ≤ i ≤ n − 1). corresponding to xi g(x) in V n [x]. Then Hg(i)

9.3 Classification and properties of cyclic codes

151

Proof This follows from the fact that g(x)h(x) = xn − 1 (see Exercise 9.12).

Theorem 9.13 The matrix H is a check matrix for the cyclic code C = g(x), and the dimension of C is k.

Proof A word c in C corresponds to c(x) = f (x)g(x) in V n [x]. If f (x) = f0 + f1 x + · · · + fn−1 xn−1 , the product f (x)g(x) can be written as c(x) = f0 g(x) + f1 xg(x) + · · · + fn−1 xn−1 g(x), and so it follows that c is given by c = f0 g + f1 g(1) + · · · + fn−1 g(n−1) . ′ By Lemma 9.12, Hg(i) = 0′ for 0 ≤ i ≤ n − 1, and so Hc′ = 0′ . Finally, suppose y = y0 y1 · · · yn−1 is in C. Since h0 = 1, the equation corresponding to the first row of Hy′ = 0′ is

yk = hk y0 + hk−1 y1 + · · · + h1 yk−1 . So if the values of y0 , y1 , . . . , yk−1 are given, the value of yk is determined. The equation corresponding to the second row determines yk+1 in terms of y1 , y2 , . . . , yk , and so on. Since there are 2k possible values for y0 , y1 , . . . , yk−1 , we have |C| = 2k , as claimed. The foregoing results imply that, in order to find all cyclic codes with wordlength n, it suffices to find the irreducible factors of xn − 1 in F2 [x]. It is tedious to find these factors ‘by hand’, but computer algebra systems such as MAPLE can do it very quickly.

Example 9.14 Given that there are three irreducible factors of x7 − 1 in F2 [x]: x7 − 1 = (1 + x)(1 + x + x3 )(1 + x2 + x3 ),

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9. Algebraic coding theory

what are the possibilities for a cyclic code with word-length 7? Solution By combining the three irreducible factors in all possible ways we can obtain 23 = 8 divisors of x7 − 1 in F2 [x]. They are the trivial divisors 1 and x7 − 1, together with 1 + x,

1 + x + x3 ,

1 + x2 + x3 ,

(1 + x)(1 + x2 + x3 ),

(1 + x)(1 + x + x3 ),

(1 + x + x3 )(1 + x2 + x3 ).

Each of the divisors generates a cyclic code, and (by Theorems 9.10 and 9.11) these are the only cyclic codes of length 7. Clearly 1 is the code in which every word is a codeword, and x7 − 1 = 0 is the code in which the only codeword is 0. The other codes are more interesting. For example, let C be the code with canonical generator g(x) = 1 + x + x3 , so that h(x) = (1 + x)(1 + x2 + x3 ) = 1 + x + x2 + x4 and h∗ = 1011100. It follows from Theorem and a check matrix for C is ⎛ 1 0 1 1 1 ⎝0 1 0 1 1 0 0 1 0 1

9.14 that the dimension of C is 4, 0 1 1

⎞ 0 0⎠. 1

The seven columns of this matrix are distinct and nonzero, so the code C is equivalent to the Hamming code in F2 7 .

EXERCISES 9.8. Write down the factors x5 −1 in F2 [x], and hence determine all cyclic codes of length 5. 9.9. Complete the classification of cyclic codes of length 7, along the lines of Example 9.14. 9.10. Show that when n is odd, x − 1 occurs exactly once as a factor of xn − 1 in F2 [x]. (It is worth remembering that x − 1 and 1 + x are the same in F2 [x].) 9.11. The factorization of x15 − 1 in F2 [x] is

x15 −1 = (1+x)(1+x+x2 )(1+x+x4 )(1+x3 +x4 )(1+x+x2 +x3 +x4 ). Using this result, find a canonical generator for a cyclic code equivalent to the Hamming code of length 15.

9.12. Write down the equations for the coefficients of g(x) and h(x) that result from the fact that g(x)h(x) = xn − 1, and hence prove Lemma 9.12.

9.4 Codes that can correct more than one error

153

9.4 Codes that can correct more than one error The following generalization of Theorem 8.10 is the starting point for the construction of families of codes with arbitrarily large minimum distance.

Theorem 9.15 Let H be a check matrix for a binary code C. Then the minimum distance of C is the equal to the minimum number of linearly dependent columns of H.

Proof Suppose the minimum distance of C is δ. Then there is a word c = c0 c1 · · · cn−1 in C with weight δ (Lemma 8.3). Let h′i denote column i of H. The equation Hc′ = 0′ can be written as c0 h′0 + c1 h′1 + · · · cn−1 h′n−1 = 0′ . In this equation, δ of the coefficients ci are equal to 1 and the others are 0. Hence the equation says that δ columns of H are linearly dependent. Conversely, if a set of columns of H is linearly dependent, the word in which the ith bit is 1 when the ith column is in the set is a codeword. Stated positively, the theorem says that a check matrix in which every set of r columns is linearly independent defines a code with minimum distance r + 1, at least. The following lemma shows that a particular kind of matrix has this property, and it will be used to justify the construction given in the next section. Note that the result holds in any field.

Lemma 9.16 Suppose F is a field and a ∈ F is such that a0 = 1, a, a2 , a3 , . . . , an−1 are distinct non-zero elements of F . Let A be the r × n matrix such that the entry in row i and column j is aij , where the rows are labelled i = 1, 2, . . . , r and the columns are labelled j = 0, 1, . . . , n − 1. Then any r columns of A are linearly independent over F .

Proof Let AS be the r × r submatrix of A comprising the set S of columns labelled

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9. Algebraic coding theory

s1 , s2 , . . . , sr :

as1 ⎜ a2s1 AS = ⎜ ⎝ . ars1 ⎛

as2 a2s2 . rs2 a

. . . .

. . . .

⎞ . asr . a2sr ⎟ ⎟. . . ⎠ . arsr

We must show that the columns of AS are linearly independent, which is equivalent to showing that the determinant of AS is not zero. Since ask is a factor of every term in column sk , det AS has factors as1 , as2 , . . . , asr . Also, subtracting column sk from column sℓ produces a column in which every term has asℓ − ask as a factor. The product of these factors yields a polynomial in a of degree s1 + 2s2 + · · · + rsr . On the other hand, the highest power of a that occurs in the expansion of the determinant is the diagonal term as1 a2s2 · · · arsr , so there are no other factors. Hence  (asℓ − ask ), det AS = ±as1 +s2 +···+sr

where the product is taken over all pairs such that sℓ > sk . Since it is given that asℓ = ask , it follows that det AS = 0.

EXERCISES 9.13. Without using the argument given in the proof of Lemma 9.16, verify that ⎛ 2 ⎞ a a5 a7 det ⎝ a4 a10 a14 ⎠ = a14 (a5 − a2 )(a7 − a2 )(a7 − a5 ). a6 a15 a21 9.14. Explain why Theorem 8.10 is a special case of Theorem 9.15. 9.15. Suppose we wish to assign ID numbers in the form of binary words of length n to a set of 16 people, so that the words form a 2-errorcorrecting linear code – that is a linear code with parameters (n, 4, 5). Show that we shall require n to be at least 10. 9.16. Given the conditions stated in the previous exercise, there is in fact no solution with n = 10. [You are not asked to prove this.] However, there is a solution with n = 11, which we are going to construct. Let A be the linear code with parameters (6, 3, 3) discussed in Example 8.6 and Section 8.3), and let B = {000000, 111111} be the code with parameters (6, 1, 6). Let C ⊆ F2 12 be the code formed by words of the form [a | a + b] a ∈ A, b ∈ B.

9.5 Definition of a family of BCH codes

155

Show that C is a linear code with parameters (12, 4, 6) and construct a linear code with parameters (11, 4, 5) by making a suitable modification to C. 9.17. Is the code constructed in the previous exercise a cyclic code?

9.5 Definition of a family of BCH codes Families of codes that can correct any number of errors were discovered by Bose and Ray-Chaudhuri, and (independently) by Hocqhenghem, in 1959-60. These codes are known as BCH codes. Here we shall consider only one case: a family of binary cyclic codes such that, for given integers m and t, the parameters (n, k, δ) satisfy n = 2m − 1,

k ≥ n − tm,

δ ≥ 2t + 1.

Since the BCH codes are cyclic codes, we begin with the factorization of x − 1 into irreducible polynomials, as in Section 9.3. Suppose it is n

xn − 1 = (x − 1)f1 (x)f2 (x) . . . fℓ (x)

in F2 [x].

There is only one possible linear factor x − 1, and when n is odd it occurs only once (Exercise 9.10), so the polynomials f1 (x), . . . , fℓ (x) all have degree greater than 1. However, if we extend the field then xn − 1 can be split completely into linear factors – in other words, it has n roots. (This procedure is analogous to extending the real field to the complex field, in order to obtain n roots for any polynomial of degree n with real coefficients.) Our definition of BCH codes will involve the construction of a field E containing F2 , and an element α ∈ E such that α is a root of the equation xn = 1 in E. Any such α has the property that α, α2 , α3 , . . . , αn−1 are roots of xn = 1. We shall require that these elements of E are distinct, and in that case we shall say that α is a primitive root. The construction of α and E will be given shortly, but for the time being we shall proceed on the assumption that it can be done. This means that we can write xn − 1 = (x − 1)(x − α)(x − α2 ) . . . (x − αn−1 )

in E[x].

Lemma 9.17 If α is a primitive root, each of the terms x − αi (1 ≤ i ≤ n − 1) is a factor of exactly one of the polynomials fj (x) (j = 1, 2, . . . , ℓ) in E[x].

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9. Algebraic coding theory

Proof Since the coefficients of fj (x) are in F2 , which is contained in E, we may consider fj (x) as an element of E[x]. Unique factorization holds in E[x], and it follows by comparing the two factorizations of xn − 1 that fj (x) must be a product of terms of the form x − αi . Since the total degree is the same both cases, a given factor x − αi can occur in only one fj (x). For i = 1, 2, . . . , n − 1 let mi (x) be the unique irreducible factor fj (x) of xn − 1 in F2 [x] for which x − αi is a factor of fj (x) in E[x]. Equivalently, mi (x) = fj (x) when fj (x) has αi as a root.

Definition 9.18 (BCH code, designed distance) Given an odd integer n, let the polynomials mi (x) ∈ F2 [x] be as above, and define g(x) to be the least common multiple of m1 (x), m2 (x), . . . , mdd−1 (x). Then we say that the cyclic code in F2 n with canonical generator g(x) is a BCH code with designed distance dd. The reason for taking the least common multiple is that the polynomials mi (x) may not all be distinct. For example, it is easy to show (Exercise 9.18) that for any f (x) ∈ F2 [x] we have f (x)2 = f (x2 ). It follows that if αi is a root of fj (x) then so is α2i . Hence the polynomials mi (x) and m2i (x) are the same, and in the definition of g(x) we need only consider the polynomials mi (x) with i odd. When dd is an odd number, the definition of g(x) becomes g(x) = lcm{m1 (x), m3 (x), . . . , mdd−2 (x)}.

We must now address the problem of constructing the primitive root α and the field E. Consider the case n = 7, when the irreducible factors of x7 − 1 are x7 − 1 = (1 + x)(1 + x + x3 )(1 + x2 + x3 ). Any root of x7 − 1 must be a root of one of the factors. The only root of 1 + x is 1, which is clearly not primitive. If α is a root of 1 + x + x3 , the equation 1 + α + α3 = 0 can be written as α3 = 1 + α. Using this equation, any power

9.5 Definition of a family of BCH codes

157

of α can be reduced to a polynomial in α with degree not greater than 2. Thus α3 α4 α5 α6 α7

=1+α = α(1 + α) = α + α2 = α(α + α2 ) = α2 + α3 = 1 + α + α2 = α(1 + α + α2 ) = 1 + α2 = α(1 + α2 ) = 1.

Using the correspondence c0 + c1 α + c2 α2

←→

c0 c1 c2

we have α α5

= 010, = 111,

α2 α6

= 001, = 101,

α3 α7

= 110, = 100.

α4

= 011

We have represented the 7 powers of α as the 7 distinct nonzero elements of F2 3 . This means that we can take E to be F2 3 , with the appropriate operation of multiplication: the product of a0 a1 a2 and b0 b1 b2 is obtained by multiplying the polynomials a0 + a1 α + a2 α2 and b0 + b1 α + b2 α2 , and using the relation 1 + α + α3 = 0 to reduce the answer to a polynomial of the same form. The same method works whenever n = 2m − 1: it is always possible to find one of the irreducible factors of xn − 1 that defines a primitive root α. The reader who has studied Galois fields will recognise the construction of GF (2m ). To summarize: when n = 2m − 1 the method described above can always be used to construct a field E with 2m elements, and a primitive root α ∈ E. Furthermore, as we shall see in the next section, the same method produces an explicit check matrix for a BCH code with n = 2m − 1 and a given value of dd. Here is a simple example, in which the check matrix is obtained directly.

Example 9.19 Take n = 7 and α to be a root of 1 + x + x3 . Find the canonical generator for the BCH code of designed distance dd = 7, and show that the code has minimum distance δ = 7. Solution Since dd = 7, the canonical generator is the least common multiple of m1 (x), m3 (x) and m5 (x). It is given that α is a root of 1 + x + x3 , so m1 (x) = 1 + x + x3 . In order to determine m3 (x) and m5 (x) we need to find the factors that have α3 and α5 as roots. Using the representation as elements of F2 3 obtained above we have 1 + (α3 )2 + (α3 )3 = 1 + α6 + α2 = 100 + 101 + 001 = 000,

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9. Algebraic coding theory

1 + (α5 )2 + (α5 )3 = 1 + α3 + α = 100 + 110 + 010 = 000. In other words, both α3 and α5 are roots of 1 + x2 + x3 , and m3 (x) = m5 (x) = 1 + x2 + x3 . Hence g(x) = lcm{m1 (x), m3 (x), m5 (x)} = (1 + x + x3 )(1 + x2 + x3 ). Following the rules given in Section 9.3 we find h(x) = 1 + x, so the dimension is k = 1. The first row of the check matrix is h∗ = 1100000 and the other rows are the cyclic shifts 0110000, . . ., 0000011. The resulting equations imply that x1 x2 . . . x7 is a codeword if and only if x1 = x2 = · · · x7 , so the code is {0000000, 1111111}. Clearly, this code has minimum distance 7.

EXERCISES 9.18. Prove that f (x)2 = f (x2 ) for any polynomial f (x) ∈ F2 [x]. Hence determine all the polynomials mi (x), 1 ≤ i ≤ 6 when n = 7 and the primitive root is a root of 1 + x + x3 . 9.19. Factorize x3 − 1 in F2 [x], and hence determine the binary BCH code in F2 3 with dd = 3. (Refer to Exercise 9.5.) 9.20. Show that a root of 1 + x2 + x3 can be taken as a primitive root of x7 − 1.

9.6 Properties of the BCH codes Suppose we know a primitive root α of xn − 1. The following lemma enables us to construct a check matrix for the corresponding BCH code without having to calculate the canonical generator explicitly.

Lemma 9.20 Let n = 2m − 1. If c = c0 c1 . . . cn−1 is a codeword in a binary BCH code of designed distance dd = 2t + 1, constructed using a primitive root α, then H ♭ c′ = 0′ , where H ♭ is the 2t × n matrix over E given by ⎞ ⎛ αn−1 1 α α2 . . . ⎜ 1 α2 α4 . . . α2(n−1) ⎟ ⎟. H♭ = ⎜ ⎝. ⎠ . . . . 2t 4t 2t(n−1) 1 α α ... α

9.6 Properties of the BCH codes

159

Proof The polynomial c(x) representing the codeword c is a multiple of g(x) which, by Definition 9.18, is a multiple of m1 (x), m2 (x), . . . , m2t (x). Since mi (αi ) = 0 it follows that c(αi ) = 0, that is c0 + c1 αi + c2 α2i + . . . + cn−1 αi(n−1) = 0

(i = 1, 2, . . . , 2t).

These equations are equivalent to the matrix equation H ♭ c′ = 0′ .

Theorem 9.21 Let C be a binary BCH code with word-length n = 2m −1 and designed distance dd = 2t + 1. Then the dimension k and the minimum distance δ of C satisfy k ≥ n − tm,

δ ≥ dd.

Proof Suppose we have identified a primitive root α of xn − 1 and represented the powers αi (1 ≤ i ≤ n) as elements of F2 m . If we replace the entries of the matrix H ♭ by the corresponding words in F2 m (as column vectors), we obtain an 2tm × n matrix H over F2 . The equations c0 + c1 αi + c2 α2i + . . . + cn−1 αi(n−1) = 0

(i = 1, 2, . . . , 2t),

now correspond to Hc′ = 0′ , so H is a check matrix for C. The 2tm rows of H are not all needed, since some of the polynomials m1 (x), m2 (x), . . . , m2t (x) coincide. Indeed, we know that mi (x) and m2i (x) are certainly the same. Thus the number s of distinct polynomials is at most t. It follows that H can be be replaced by an sm × n matrix, and the dimension of the code is k = n − sm ≥ n − tm. Finally, Lemma 9.16 shows that any 2t columns of H ♭ are linearly independent, so the same is true for H, and the minimum distance of C is such that δ ≥ 2t + 1 = dd.

Example 9.22 The factorization of x15 − 1 in F2 [x] is x15 − 1 = (1 + x)(1 + x + x2 )(1 + x + x4 )(1 + x3 + x4 )(1 + x + x2 + x3 + x4 ). Show that a root α of 1 + x + x4 can be chosen as primitive root, and hence write down a check matrix for the BCH code with n = 15 and dd = 5.

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9. Algebraic coding theory

Solution Since α4 = 1 + α, any power of α can be reduced to a polynomial in α with degree at most 3. Using the correspondence c0 + c1 α + c2 α2 + c3 α3 ←→ c0 c1 c2 c3 we find that α α5 α9 α13

= 0100, = 0110, = 0101, = 1011,

α2 α6 α10 α14

= 0010, = 0011, = 1110, = 1001,

α3 α7 α11 α15

= 0001, = 1101, = 0111, = 1000.

α4 α8 α12

= 1100 = 1010, = 1111,

Since these vectors are all distinct, we conclude that α is a primitive root. In the case dd = 5, g(x) is the lcm of m1 (x), m2 (x), m3 (x). Since m2 (x) = m1 (x) a check matrix for the code is the 8 × 15 matrix over F2 ,   1 α α2 α3 α4 α5 α6 . . . α14 H= , 1 α3 α6 α9 α12 1 α3 . . . α12 where αi stands for the corresponding column vector, as listed above. According to Theorem 9.21, the code constructed above has δ ≥ dd = 5, and so it is a 2-error-correcting code. There is a simple error-correction procedure. Suppose a word z = z0 z1 . . . z14 is received. As usual, we begin by calculating the syndrome s, say s′ = Hz′ = [x y]′ , where x and y are 4-vectors. • If x = y = 0000, then z is a codeword, and there are no bit-errors. • If, for some i, x = αi and y = α3i , the syndrome is the ith column of H, and there is an error in the ith bit of z. • If it is possible to find i and j satisfying the equations x = αi + αj ,

y = α3i + α3j ,

then the syndrome is the sum of the ith and jth columns of H, and z has errors in the ith and jth bits. So we must solve these equations for i and j. If the equations have no solution, it must be assumed more than two errors have been made.

Example 9.23 Suppose two errors are made in transmission and the syndrome of the received word z is s′ = Hz′ = 1100 0110′. Which bits are in error?

9.6 Properties of the BCH codes

161

Solution In this case, x = 1100 and y = 0110. If errors have been made in the ith and jth bits, we have to find i and j such that αi + αj = 1100

α3i + α3j = 0110.

One way to do this is to make a list of the pairs (αi , αj ) for which the first equation is satisfied: (1, α),

(α, 1),

(α2 , α10 ) (α3 , α7 ),

...

.

Now we can check whether the corresponding pairs satisfy the second equation. 1 + α3 = 1000 + 0001 = 1001, α3 + 1 = 0001 + 1000 = 1001, α6 + 1 = 0011 + 1000 = 1011, α9 + α6 = 0101 + 0011 = 0110. The conclusion is that bits labelled 3 and 7 are in error, so the intended codeword was c = z + 000100010000000.

EXERCISES 9.21. Compare the information rate and error-correction properties of the BCH code constructed in Example 9.22 with those of the Hamming code with the same word-length. 9.22. Suppose that the BCH code in Example 9.22 is being used, and a word with syndrome 10101001 is received. Assuming that two biterrors have been made, which bits should be corrected? 9.23. Investigate the binary BCH codes with n = 15 and dd = 7, 9, 11, 13, 15, taking a root of 1 + x + x4 as a primitive root as in Example 9.22.

Further reading for Chapter 9 Hamming’s construction of perfect codes was published in 1950 [9.4]. Around the same time Marcel Golay discovered two very remarkable perfect codes that can correct more than one error [9.2], and it is now known that these are the only perfect codes with that property. The Golay codes have many links with other interesting structures in group theory, design theory, and geometry.

162

9. Algebraic coding theory

One of the original motivations for studying cyclic codes was that the shift operations can be implemented by a device known as a shift-register. The general results are mainly due to E. Prange [9.6] and W.W. Peterson [9.5]. Many families of cyclic codes have been constructed, and they have found numerous practical applications, ranging from space exploration to data processing. The BCH codes were first published in 1960 [9.1], and there is now an extensive literature about them: a good introduction is given by Pretzel [9.7]. Important advances are still being made in this area – see, for example, Guruswami and Sudan [9.3]. 9.1 R.C. Bose and D.K. Ray-Chaudhuri. On a class of error correcting codes. Info. and Control 3 (1960) 68-79, 279-290. 9.2 M.J.E. Golay. Notes on digital coding. Proc. IEEE. 37 (1949) 657. 9.3 V. Guruswami and M. Sudan. Improved decoding of Reed-Solomon and algebraic-geometric codes. IEEE Trans. Info. Theory 45 (1999) 1757-1767. 9.4 R.W. Hamming. Error detecting and error correcting codes. Bell System Tech. J. 29 (1950) 147-160. 9.5 W.W. Peterson and E.J. Weldon. Error-Correcting Codes. MIT Press, Cambridge, Mass. (1972). 9.6 E. Prange. The use of information sets in decoding cyclic codes. IEEE Trans. Info. Theory. 8 (1962) 55-59. 9.7 O. Pretzel. Error-Correcting Codes and Finite Fields. Oxford University Press, Oxford (1992).

10 Coding natural languages

10.1 Natural languages as sources Thus far we have studied coding for the purposes of economy and reliability. The third main purpose, security, involves (among other things) the secrecy of messages written in a natural language. For that reason, the complex properties of natural languages play an important part in cryptography. We begin by discussing how a natural language, specifically English, may be considered as a source in the sense of coding theory. Our mathematical model of this source will be referred to as english. It is based on the alphabet A with 27 symbols, the letters A, B, C, . . ., Z, and the space, denoted by ⊔. In english there is no distinction between upper and lower case letters, and there are no punctuation marks. For example, the text Rhett: Frankly my dear, I don’t give a damn! is rendered as the following string in the english alphabet. RHETT⊔FRANKLY⊔MY⊔DEAR⊔I⊔DONT⊔GIVE⊔A⊔DAMN Although we use the simplified alphabet A, we shall try to ensure that english has statistical properties that resemble the real English language. N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 10, 

164

10. Coding natural languages

The fundamental fact is the observation that the frequencies of the symbols are found to be fairly constant over a wide range of texts, ranging from Jane Austen to the Electronic Journal of Analytical Philosophy. Typical values of these frequencies, expressed as the number of occurrences per 10000 symbols, are shown in Figure 10.1. ⊔ A 1753 724

B 75

C D E F G H 231 319 1042 205 126 419

I 641

K 53

L M 320 145

R 557

J 9

S T U 668 842 195

V 91

N 554 W 172

O P 551 148 X 9

Y 133

Q 6 Z 12

Figure 10.1 A frequency table for symbols in english This frequency table defines a probability distribution p on A. But clearly this table does not tell the whole story about english. For example, given that one symbol is the letter Q, there is a very high probability that the next symbol is U. In other words, it cannot be assumed that english is a memoryless source. In cryptography, a pair of consecutive symbols is known as a digram, and the relative frequencies of digrams are significant. For example, it is observed that the digram UP occurs more frequently that the digram UE. Part of a frequency table for digrams is shown in Figure 10.2. In this table the number in row B and column E is the number of occurrences of the digram BE in a typical piece of text with 10,000 symbols. ⊔ A B C D E ⊔ − 212 57 86 69 28 A 41 − 3 38 22 − B − 3 − − − 35 C − 41 − − − 44 D 135 22 − − − 54 E 365 57 − 16 73 24 ... ... ... ... ... ... ...

... ... ... ... ... ... ... ...

Figure 10.2 Part of a frequency table for digrams in english In mathematical terms, the frequency table for digrams defines a probability distribution p2 on A2 . It is clear that for any ordered pair of symbols ij, the

10.2 The uncertainty of english

165

probability p2 (ij) is not equal to p(i)p(j). For example, according to the table given above p(A) = 0.0724 and p(B) = 0.0075,

so p(A)p(B) ≈ 0.0006,

whereas p2 (AB) is approximately 0.0003.

EXERCISES 10.1. If a frequency table for digrams is given, how can a frequency table for the individual symbols be obtained? 10.2. On the basis of the frequency table shown in Figure 10.1, estimate the average length of a word in english.

10.2 The uncertainty of english Recall that a stationary source has the property that the probability of finding any given sequence of consecutive symbols is the same at all points of the emitted stream. If we think of a novel as a stream of symbols emitted by the source called english, the stationary property means that the probability of finding BOOT (for example) on page 1 is the same as finding it on page 99, or on any other page. Formally, we require that for each n ≥ 1 there is a probability distribution pn on the set of n-tuples An . If the stream emitted by the source is represented by the sequence of random variables ξ1 ξ2 ξ3 · · · , then for any k ≥ 1 and for any n-tuple of symbols x1 x2 . . . xn ∈ An we require that Pr(ξk+1 = x1 , ξk+2 = x2 , . . . , ξk+n = xn ) = pn (x1 x2 · · · xn ). It is reasonable to assume that the statistical properties of english, and any other natural language, can be modelled by representing it as a stationary source. For the time being we shall focus on the implications of this assumption, remembering that it may not cover all the features of english. In particular, we recall that there is good definition of the entropy of a stationary source (Definition 4.8). In the case of a natural language we shall be concerned (initially at least) with coding a message in the source alphabet by a string of symbols in the same alphabet. For that reason it is appropriate to measure the entropy in terms of logarithms to base b, where b is the number of symbols in the alphabet. For

166

10. Coding natural languages

example, in english b = 27. In order to emphasize this point, we shall use the word uncertainty in this context.

Definition 10.1 (Uncertainty of a natural language) Suppose we regard a natural language with an alphabet of size b as a stationary source represented by distributions pn (n ≥ 1). Let Un = Hb (pn )/n. Then the uncertainty of the language is defined to be U = inf Un . n∈N

Our choice of units means that the number Un =

H2 (pn ) Hb (pn ) = n n log2 b

lies between 0 and 1. It can be thought of as the average uncertainty per symbol, when the source is considered as emitting a stream of blocks of size n.

Example 10.2 A language has three symbols a, b, c, and the frequency table for digrams is

a b c

a b c 0.22 0.08 0.10 . 0.10 0.16 0.04 0.08 0.06 0.16

If the language is considered as a stationary source, what are the values of U1 and U2 ? Solution we have

Using the the addition formula for p1 in terms of p2 (Section 4.3)

p1 (a) = 0.22 + 0.08 + 0.10 = 0.4, p1 (b) = 0.10 + 0.16 + 0.04 = 0.3, p1 (c) = 0.08 + 0.06 + 0.16 = 0.3. Hence U1 = H3 (p1 ) = 0.4 log3 (1/0.4) + 0.3 log3 (1/0.3) + 0.3 log3 (1/0.3) ≈ 0.9912.

10.2 The uncertainty of english

167

For U2 we have U2 =

 1 H3 (p2 ) = 0.5 0.22 log3 (1/0.22) + 0.08 log3 (1/0.08) + · · · 2  + · · · + 0.16 log3 (1/0.16) ≈ 0.9474.

The fact that U2 < U1 reflects the fact that the uncertainty is less when the relationship between consecutive symbols is taken into account. On the assumption that english is a stationary source, we can attempt to estimate its uncertainty by using experimental data. We start with the firstorder approximation, consisting of the distribution p1 on A, as measured by the frequency table (Figure 10.1). This results in the estimate  p1 (i) log27 (1/p1 (i)), U1 = H27 (p1 ) = i∈A

which works out at about 0.85. The second-order approximation is determined by the frequency table for digrams, which defines a distribution p2 on the set A2 . This gives the estimate U2 =

1 1  2 H27 (p2 ) = p (ij) log27 (1/p2 (ij)). 2 2 2 ij∈A

After a rather long calculation, a value of about 0.70 is obtained. Unfortunately, computing reliable estimates Un for larger values of n is very difficult, because there are 27n individual contributions to the entropy, and most of them are very small. Nevertheless, it is reasonable to expect that, as longer sequences are taken into account, the estimate of uncertainty per symbol will decrease, because common words such as AND, THE, and THIS will contribute significantly to the results. On the other hand, at this level it is hard to justify the claim that every human author represents the common source that we should like to call english. For example, some authors avoid the use of certain four-letter words, while others are less constrained. For the record, we simply note that the conventional assumption, based on extensive calculations, is that U = inf Un ≈ 0.3. n∈N

Roughly speaking, if a long piece of text is given, then we can predict what comes next with about 70% certainty. In the next section we shall look at this result from a different point of view.

168

10. Coding natural languages

EXERCISES 10.3. The language footish is mainly used by professional footballers. The language uses a set of three symbols, {∗ ! ?}. An example of a string recently emitted is as follows. ?! ∗ ∗?!! ∗ ∗??!! ∗ ∗?! ∗ ∗? Estimate the uncertainty of the first-order and second-order approximations to footish. 10.4. What is the value of U1 for a language in which all the symbols are equally probable? 10.5. Consider a language with two symbols x, y, such that the frequency table for digrams is as follows.

x y

x y 0.3 0.2 0.2 0.3

Show that U1 = 1 but U < 1 for this language.

10.3 Redundancy and meaning There are good reasons for believing that the property of being a stationary source does not capture all the features of a natural language. In addition to its non-trivial statistical properties, a natural language has another fundamental property, which we may loosely refer to as meaning. Although it is plain that the function of natural language is to convey meaningful messages, it is not easy to express this idea in mathematical terms. One difficulty is that it is quite possible to generate strings of symbols that make no sense, although they have statistical properties very similar to the source that we have called english. Suppose we take a book, assumed to be a typical output of english. If we rearrange the pages of the book, and the sentences on each page, we shall have an output that is virtually indistinguishable (in statistical terms) from the original book. But clearly, it makes no sense. One characteristic property of a meaningful message is that it is possible to shorten it without destroying the meaning. At a very basic level that is the reason why we can use abbreviations, such as LSE and USA. Similarly, omitting some letters from a meaningful message will not prevent it being understood.

10.3 Redundancy and meaning

169

For example, 7 of the original 29 symbols have been omitted from the message IT⊔I⊔GINGT⊔RAI⊔TOMORW but the meaning has not been lost. This property of a natural language is known as redundancy. We now give a heuristic argument, due to Shannon, that links the redundancy of a language with its uncertainty. Consider the set of meaningful messages of length n emitted by the source that we call english. Suppose that it has been found by experiment that the proportion of symbols that can be omitted from these messages without destroying the meaning is fn . The process of removing nfn symbols can be thought of as encoding the message of length n by a message in the same alphabet, but with length ℓn = n(1 − fn ). Now the coding theorem for stationary sources (Theorem 4.11) says that the optimum value of ℓn /n is close to U, the uncertainty of the source. Thus it reasonable to assume that inf fn = inf (1 − ℓn /n) = 1 − U.

n∈N

n∈N

Shannon’s argument suggests that it is reasonable to make the following definition.

Definition 10.3 (Redundancy) The redundancy of a natural language with b symbols is defined to be R = 1 − U, where U is the uncertainty as in Definition 10.1. There are several ways of looking at this definition. One of them is to regard it as an alternative means of calculating U, using experimental results on redundancy. Most experiments suggest that well over half of the letters in a long message can be omitted, although naturally the results vary according to the rule that is used to suppress the letters. This observation is consistent with the estimate U ≈ 0.3 given in the previous section. In summary, there are several ways of estimating the uncertainty and redundancy of a natural language such as english. However, neither linguistic theorists nor mathematicians have succeeded in formulating a theory that captures all the subtleties of natural language, and so numerical estimates are very imprecise. More important is the fact that the concepts of redundancy and meaning are highly relevant to the practical aspects of cryptography, as we shall see in the next section.

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EXERCISES 10.6. Use the frequency table (Figure 10.1) to estimate approximately the percentage of a typical message in english that consists of vowels and spaces. 10.7. Reconstruct the following english messages, in which the vowels and and spaces have been removed. LTSFPPLSPPRTMNCHSTRNTD

DNTCRYFRMRGNTN

10.8. Our university administration has been told that the redundancy of language is at least 50%. In their unceasing quest for efficiency, they have applied this information in a simplistic way by sending the following message to all staff. What was their rule for suppressing symbols, and what is the intended message? PES RMME T CMLT YU RPRS BFR TE ED O TR. 10.9. The widespread use of mobile phones for sending text messages has led to a new language, which we may call textish. Discuss the relationship between textish and english. Which language has the greater uncertainty?

10.4 Introduction to cryptography We are now ready to discuss the traditional aspect of cryptography: the study of coding for the purpose of transmitting secret messages written in a natural language. Traditionally, secrecy was required mainly in diplomatic and military communications, but nowadays it plays an important part in our everyday lives, for example, in managing our financial affairs. Partly because it has a long history, cryptography has developed its own terminology. We shall begin with a brief description of a few of the methods that have been used in the past, and by doing so we shall be able to introduce most of the special terminology of the subject. We shall also encounter many of the peculiar problems that arise in this area. One of the oldest cryptographic systems is said to have been used by Julius Caesar over two thousand years ago. It was discussed briefly in Section 1.6 of this book. For a message emitted by the idealized source that we have called english, the system is as follows. The sender chooses a number k between 1 and 25 and replaces each letter by the one that is k places later, in alphabetical order, with the obvious rule for letters at the end of the alphabet. In order to

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171

simplify the explanation we shall assume that the space ⊔ is not changed. (In practice the aim is obviously not to make things simple, and a different rule would be used.) For example, when k = 5 the letters are replaced as follows. ⊔ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ⊔ F G H I J K L M N O P Q R S T U V W X Y Z A B C D E Using this rule, the message SEE⊔YOU⊔TOMORROW

becomes

XJJ⊔DTZ⊔YTRTWWTB

.

In cryptography this process is known as encryption, and the number k is the key (k = 5 in the example above). In mathematical terms, encryption is simply coding, with the additional feature that the coding function Ek depends upon a parameter k. For example, in our version of Caesar’s system Ek is the function A → A defined by Ek (x) = [x + k] (x = ⊔),

Ek (⊔) = ⊔,

where [x + k] stands for the symbol k places after x, with the obvious rule for the letters at the end of the alphabet. The function Ek is extended so that it acts on messages (strings of symbols) by concatenation: for example Ek (CAT) = Ek (C)Ek (A)Ek (T). As usual, we require that the extended Ek is an injection, so that the coding is uniquely decodable. (Definition 1.13). The general situation is as follows.

Definition 10.4 (Encryption functions) Let M and C be sets of messages, and let K be a set. Suppose that for each k ∈ K there is an injective function Ek : M → C. Then we say that {Ek } is a set of encryption functions, and an element k ∈ K is called a key. In Caesar’s system M and C are both taken to be the set A∗ of strings of symbols in A, and K is the set of numbers in the range 1 ≤ k ≤ 25. The main lesson from the history of cryptography is that it is futile to try to conceal the method of encryption, that is, the form of the functions Ek . If a number of people wish to communicate securely, they cannot hope to conceal the system that they are using. The security of the system must depend only on the specific value they agree to assign to the key k. Nowadays it is customary to give human names, Alice, Bob, Eve, to the participants in security procedures. (In practice, the participants are machines of various kinds.) Roughly speaking, Alice is the sender of a message, Bob is the

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receiver, and Eve is an adversary who is trying to intervene in some way. Within this framework we can distinguish several requirements that come under the general heading of security. The most obvious requirement is Confidentiality: a message from Alice to Bob should not be understood by Eve. This is the requirement that we shall consider in this chapter. However, modern cryptography is also concerned with other aspects of security, such as Authenticity, Integrity, and Non-repudiation. We shall discuss these matters in Chapter 14. Let us assume that Alice and Bob have agreed on a method of encryption and the value of the key k, so that they both know the encryption function Ek . The message that Alice wishes to send Bob is known as plaintext, and for the time being we shall assume that is expressed in a natural language, such as english. Alice uses the function Ek to transform the plaintext into coded form, known as ciphertext, and sends this to Bob. So if Alice and Bob have agreed to use the Caesar system with k = 5, and the plaintext is SEE⊔YOU⊔TOMORROW then Bob will receive the ciphertext XJJ⊔DTZ⊔YTRTWWTB

.

In this case it is fairly obvious how Bob should retrieve the plaintext, but it is helpful to consider the situation in more general terms. Suppose we are given a set of encryption functions Ek : M → C. According to Definition 10.4 each Ek is an injective function. This means that if Ek (m) = c, then m is the unique plaintext that Ek encrypts as c. Consequently there is a function F , defined on the image of Ek , that takes c to m: F (Ek (m)) = m

for all m ∈ M.

Formally, F is the left-inverse of Ek . In practical cryptography it is not enough to know that a left-inverse exists, it is necessary to have an explicit rule for calculating it. That is the point of the following definition.

Definition 10.5 (Decryption functions) Let {Ek } (k ∈ K) be a set of encryption functions M → C. Then the set of functions {Dℓ } (ℓ ∈ K) is a set of decryption functions for {Ek } if for each k ∈ K there is an ℓ ∈ K such that Dℓ is the left-inverse of Ek : Dℓ (Ek (m)) = m

for all m ∈ M.

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173

In the Caesar system, we take ℓ such that ℓ = −k (mod 26) and the leftinverse of Ek is Dℓ (x) = [x + ℓ] (x = ⊔),

Dℓ (⊔) = ⊔.

Since Bob is assumed to know k, he also knows ℓ, and he can decrypt the message from Alice. Note that in this case the decryption function Dℓ and the encryption function Ek have the same form, although that is not necessary in general. Consider now the situation when Eve intercepts the ciphertext XJJ⊔DTZ⊔YTRTWWTB

.

She cannot simply apply a decryption function and recover the plaintext, because she does not know which key has been used. If she wishes to obtain the plaintext, the obvious method is to try to find the key. The process of obtaining the plaintext corresponding to some ciphertext, either by finding the value of the key k or by some indirect method, is said to be breaking the system. Any method which (Eve hopes) will achieve this is known as an attack. We have already observed that Eve may be assumed to know which system is being used; consequently, if she finds the key, then she knows how to use it. For the Caesar system there is a simple attack by the method known as exhaustive search. Because the only possible values of k are 1, 2, 3, . . . , 25, it is easy to try each of them in turn. Assuming that the plaintext is a message expressed in a natural language, Eve will know when the right key is found, because the message will be ‘meaningful’.

Example 10.6 Suppose Eve intercepts the ciphertext SGZN⊔OY⊔MUUJ⊔LUX⊔EUA

.

What is the corresponding plaintext? Solution When the keys k = 1, 2, 3, . . . are tried in turn on the initial part of the ciphertext, the result is meaningless until k = 6 is reached. k k k k k k

=1 =2 =3 =4 =5 =6

RFYM⊔ QEXL⊔ PDWK⊔ OCVJ⊔ NBUI⊔ MATH⊔

··· ··· ··· ··· ··· ···

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So, it is worth trying the key k = 6 on the rest of the message. This produces a meaningful message, and the plaintext is MATH⊔IS⊔GOOD⊔FOR⊔YOU

.

EXERCISES 10.10. Find the key that has been used to produce the following ciphertext. (Here a space is represented by a blank.) VJKU KU CP GZEGNNGPV EQWTUG CPF VJG NGEVWTGT UJQWNF TGEGKXG C DKI RCA TKUG 10.11. The following ciphertext has been produced by a variant of the Caesar system in which the letters are divided into blocks of three for convenience, and the spaces between the words of the plaintext have been ignored. Find the key and the message. VJC QNV JCR LBR BXO CNW DBN ODU 10.12. The ciphertext CTGPC is the result of encrypting a meaningful word using the Caesar system. Show that there are two possible words satisfying this description.

10.5 Frequency analysis How can Alice and Bob defend against the attack by exhaustive search? One obvious idea is to increase the number of keys. Caesar’s system uses the very simple rule x → [x + k] for replacing the letters, and there are only 25 possible values of the key k. But clearly any permutation of the 26 letters can be used. There are 26! such permutations, and 26! ≈ 4 × 1027 , so Eve will require substantial resources if she wishes to try all of them in turn. In cryptography a system that simply permutes the letters of the alphabet in a specific way is known as mono-alphabetic substitution. The key is a permutation σ of the 26 letters, and the encryption function can be defined as Eσ (x) = σ(x)

(x = ⊔),

Eσ (⊔) = ⊔.

(In practice, the space would probably be permuted as well, but we shall use this definition in order to simplify the exposition.) Clearly, the left-inverse of

10.5 Frequency analysis

175

Eσ is the function Dτ that uses the inverse permutation τ = σ −1 : Dτ (x) = τ (x)

(x = ⊔),

Dτ (⊔) = ⊔.

It is often convenient to use a key that can be memorized, and this can be done by choosing a keyword. For example, if the keyword PERSONALITY is chosen, the corresponding permutation σ is defined as follows. ⊔A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ⊔P E R S O N A L I T Y B C D F G H J K M Q U V W X Z The permutation can be written compactly in cycle notation: σ = (⊔)(APG)(BEOFNDSKYXWVUQHL)(CRJTM)(I)(Z). One advantage of this notation is that is easy to write down the inverse permutation: σ −1 = (⊔)(AGP)(BLHQUVWYXYKSDNFOE)(CMTJR)(I)(Z). Although the attack by exhaustive search requires large resources, another method of attack, known as frequency analysis, is usually more effective. The method was used by Arab cryptographers in the ninth century AD. It is based on the fact that a plaintext message in a natural language will have the statistical properties discussed in Section 10.1. Furthermore, it is reasonable to assume that the text conveys a meaningful message. It follows that, in any message of reasonable length, the symbols, digrams, and so on, will occur with frequencies close to those given in the standard frequency tables. Thus if the symbol x occurs with frequency nx in the plaintext, and the encryption function is Eσ , the symbol σ(x) will occur with frequency nx in the ciphertext. Using the data provided by this observation, and the fact that the plaintext is meaningful, the ciphertext can be decrypted. In summary, the attack by frequency analysis on a piece of ciphertext produced by mono-alphabetic substitution is carried out as follows: • count the frequencies of the symbols in the ciphertext; • compare these with the standard frequencies given in the tables; • test the likely correspondences of the symbols, until a meaningful plaintext is obtained. The third step involves making and testing hypotheses, and there is no definite rule for how that should be done. The following example is based on a very crude application of the method. Obviously, more sophisticated arguments would be used in reality.

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Example 10.7 Suppose Eve intercepts some ciphertext, part of which is . . . XPYVBH VWX WROCZBVPYH . . .

.

(The space symbol ⊔ has been omitted because we are assuming, for the sake of exposition, that the spaces are unaltered.) Eve believes that Alice has sent this to Bob, using a mono-alphabetic substitution Eσ . She has counted the number of occurrences of each letter in a sample of 1000 symbols from the ciphertext, and the most frequent letters are Y Z V R H W P C X . . 97 79 64 63 60 54 47 40 37 . .

. .

.

What is the plaintext? Solution Comparing the results with the frequency table in Figure 10.1, Eve would start by assuming that the letters Y and Z represent E and T, the two most common letters in english, in that order. Similarly, the next most frequent letters, V, R, H, can be assumed to represent A, I, and S, in some order. To decide which order, Eve must test alternative hypotheses on a segment of the ciphertext, such as that displayed above. She has assumed Y → E and Z → T, and she might use the frequency table for digrams to decide that YV is more likely to represent EA than EI. Also, noting that a word is more likely to end in S than I, it would be reasonable to try V → A

R → I

H → S.

In that case the selected part of the ciphertext would be decrypted as follows: XPYVBH VWX WROCZBVPYH . . EA . S A . . . I . .T . A .E S

.

Eve will now look at the next most frequent letter, W, and assume that it stands for O or N. After a few trials she will find that the most likely correspondence is W → N. This gives XPYVBH VWX WROCZBVPYH . . EA .S AN . NI . .T . A .ES

.

Now it is easy to deduce the remaining correspondences, such as X → D, . . . , and complete the decryption. Note that, in order to find the plaintext, Eve does not have to obtain the full key (the permutation σ of A), only enough of it to give a meaningful message.

10.5 Frequency analysis

177

EXERCISES 10.13. Write down in cycle notation the permutation of A determined by the keyword REPUBLICAN. Use this key to encrypt the message HAPPY BIRTHDAY GRANNY

.

10.14. Suppose you have obtained some ciphertext, encrypted by a monoalphabetic substitution, with the spaces unaltered. The number of occurrences of the symbols that occur most frequently in a sample of about 1000 letters in the ciphertext is: A B C D E F I K 55 58 40 73 44 51 54 3

P S X Z 62 96 15 30

.

Part of the ciphertext is DA XS AF BAD DA XS DEPD CI DES KZSIDCAB

.

Find the corresponding plaintext. 10.15 Alice and Bob are exchanging messages with a mono-alphabetic substitution system. For simplicity, they wish to use exactly the same key for both encryption and decryption, while ensuring that only the space is unaltered. Write down a suitable key in cycle notation. If Eve knows that they are using a key with this property, is it possible for her to carry out an attack by exhaustive search?

Further reading for Chapter 10 ‘Natural language’ is a very broad concept, and it must be stressed that our discussion refers only to language in written form. Linguists and mathematicians have not yet succeeded in formulating rules that distinguish meaningful messages from nonsense, and for that reason we must tread carefully when we assign numerical values to concepts such as redundancy. The most successful approach so far is the experimental one suggested by Shannon [10.2], an excellent account of which is given by Welsh [10.4, Chapter 6]. The original reference to Caesar’s system is by Suetonius in the second century AD [10.3]. The ninth century Arab manuscript on frequency analysis is described in an article by Al-Kadi [10.1]. Other classical systems are described in the book by Singh listed at the end of Chapter 1 [1.3].

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10.1 I.A. Al-Kadi. The origins of cryptology: the Arab contribution. Cryptologia 16 (1992) 97-126. 10.2 C. Shannon. Prediction and entropy of printed English. Bell Syst. Tech. J. 30 (1951) 50-64. 10.3 Suetonius. Lives of the Caesars, LVI. 10.4 D. Welsh. Codes and Cryptography. Oxford University Press (1988).

11 The development of cryptography

11.1 Symmetric key cryptosystems In the previous chapter we described a framework for cryptography based on a set of plaintext messages M, a set of ciphertext messages C, and a set of keys K. For each k ∈ K there is an encryption function Ek : M → C with a left-inverse, the decryption function Dℓ . In other words, Dℓ (Ek (m)) = m

for all m ∈ M.

We shall refer to this framework as a cryptosystem. Cryptosystems can be constructed and implemented in many ways. In the mono-alphabetic substitution systems discussed in Chapter 10, the corresponding keys k and ℓ are inverse permutations, and so they are related in a very simple way. Indeed, for many centuries it was tacitly assumed that any practical cryptosystem must have a similar property. It is now clear that the assumption is unjustified, but systems with this property are still widely used, and we give them a name.

Definition 11.1 (Symmetric key cryptosystem) Suppose we have a cryptosystem in which Dℓ is the left-inverse of Ek . The system is said to be a symmetric key cryptosystem if, whenever either one of k, ℓ is known, then it is easy to calculate the other one.

N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 11, 

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The word ‘easy’ can be interpreted as indicating that the calculation can be successfully carried out on any reasonably efficient computer. We shall say more about this point in due course.

EXERCISES 11.1 Suppose we are using a mono-alphabetic substitution system, as in Section 10.5, based on the keyword DEMOCRAT. Write down the key for encryption as a permutation in cycle notation, and derive the key for decryption in the same form.

11.2 Poly-alphabetic encryption For many hundreds of years cryptographers were mainly concerned with ways of making their systems safe against the attack by frequency analysis. In a mono-alphabetic system, with a given key, the plaintext letter E (for example) is always replaced by the same letter in the ciphertext, say X. Consequently X will be distinctive in the ciphertext, because it will almost certainly be the most frequently occurring letter. This weakness can be avoided by using a system in which E is not always replaced by the same letter.

Definition 11.2 (Poly-alphabetic encryption) In a poly-alphabetic system the rule for encryption is that each plaintext letter is replaced by a letter that depends, not only on the letter itself, but also its position in the text. One simple method of poly-alphabetic encryption was known as long ago as the 16th century, and is usually known as the Vigen`ere system. It uses only the ‘Caesar’ permutations of the letters – that is the cyclic shifts of step-size k (1 ≤ k ≤ 25), denoted in the previous chapter by Ek (x) = [x + k]. As usual, for the purposes of exposition we ignore the spaces, in this case by deleting them from the plaintext altogether. In the Vigen`ere system the key is a sequence K = (k1 , k2 , . . . , km ) of numbers. If the plaintext is x1 x2 x3 . . . xi . . . , the rule for encryption is that EK (xi ) = [xi + kj ] if i ≡ j (mod m).

11.2 Poly-alphabetic encryption

181

For example, if the key has length m = 4, then x1 , x5 , x9 , . . . are encrypted with a shift of k1 , x2 , x6 , x10 , . . . are encrypted with a shift of k2 , and so on. This is obviously a symmetric key system, since decryption can be done by reversing the shifts. In practice the key K is usually expressed as a keyword, the letters of the keyword representing the relevant shifts.

Example 11.3 Suppose the keyword is CHANGE, representing the key K = (3, 8, 1, 14, 7, 5). What is the encrypted form of the following plaintext? THEPROOFOFTHEPUDDINGISINTHEEATING Solution The letters in positions 1, 7, 13, 19, 25, 31, 37 are T,O,E,N,T,I, and they are encrypted with a shift of 3, becoming W,R,H,Q,W,L. The other letters are encrypted in a similar way. The entire process can be set out as follows. T H 3 8

E 1

P R O 14 7 5

O F 3 8

O 1

F T H 14 7 5

E P 3 8

U 1

D D I 14 7 5

N 3

G ... 8 ...

W P

F

D

R N

P

T

H X

V

R

Q

O ...

Y T

A M

K N

It is clear that the Vigen`ere system has the desired effect of smoothing out the frequencies of the letters in the ciphertext. For example, suppose that a piece of english plaintext contains letters with their usual frequencies (Figure 10.1) and the keyword is CHANGE, as in the example above. Then the ciphertext letter A represents one of the letters X S Z M T V, and hence its expected frequency is 1 (9 + 668 + 12 + 145 + 842 + 91) ≈ 294. 6 Similarly, the letter Z in the ciphertext represents one of the letters W R Y L S U, so its expected frequency is 1 (172 + 557 + 133 + 320 + 668 + 195) ≈ 341. 6 These numbers are much closer than would be expected if a mono-alphabetic substitution were used. (Note that since we are ignoring spaces, which occur approximately 1753 times in every 10000 symbols, the frequency of each letter is here expressed as a proportion of 8247 symbols, rather than 10000. But this does not affect the conclusion.)

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For several centuries the Vigen`ere system was thought to be unbreakable, and was referred to as le chiffre ind´eciffrable – the undecipherable cipher. However, the system has an inherent weakness, because the choice of the shifts must be based on a definite rule, otherwise the intended receiver will not be able to decrypt. This weakness can be exploited to mount a successful attack. The basic strategy is simple. If the length m of the keyword is known, then the letters occurring in positions 1, m + 1, 2m + 1, . . . are all encrypted with the same shift k1 , and hence k1 can be found by standard frequency analysis. The same holds for the letters in positions i, m + i, 2m + i, . . ., for any i. Hence if m is known, finding the keyword itself presents no difficulty, provided that a reasonably large chunk of ciphertext is available. It was not until the 19th century that an effective method of finding m was discovered. The method has since been refined, and it is now almost automatic. However, a significant amount of calculation is required and a realistic example would require more space than is available here. But the basic idea is easy to understand. Take a piece of ciphertext c and translate it forward by ℓ places, so that the ith symbol in c is the (i + ℓ)th symbol in the new text. If ℓ is not equal to m (or a multiple of m) there will be no correlation between the two texts. But if ℓ is equal to m (or a multiple of m) then corresponding symbols have been encrypted by the same substitution, and some correlation can be expected. It turns out that a simple statistic, the index of coincidence, measures the degree of correlation. Thus, provided that a reasonable sample of ciphertext is available, the value of m will be revealed by calculating this statistic for a range of values of ℓ. The important general conclusion is that no cryptosystem can be guaranteed secure against unforeseen methods of attack.

EXERCISES 11.2. Complete the encryption of the message in Example 11.3. 11.3. The ciphertext MKWJAFQCHIUPVPONWHFDRKEFIROFEHGQMRGM is the result of encrypting a message in the Vig`enere system, using the keyword SCRAMBLE. Find the message. 11.4. Suppose Vig`enere encryption with the keyword CHANGE is applied to a piece of plaintext in which the frequencies of the letters are as in english (Figure 10.1). What is the frequency of the letters D and Q in the ciphertext? What would be the frequency of each letter (per 8247 letters) if the distribution were uniform?

11.3 The Playfair system

183

11.3 The Playfair system Although it was thought to be unbreakable, the Vigen`ere system was difficult to use. Consequently there was considerable interest in alternative methods. One such method was the Playfair system, in which the digrams are permuted, rather than the individual letters. A key for the system is derived from a 5 × 5 square containing 25 letters. (In order to make a square, J is identified with I, and spaces are ignored.) Doubleletter digrams of the form LL are not allowed, so the number of digrams available is 25 × 24 = 600. The number of keys is therefore 600!, which is enormous, and finding the key by exhaustive search is unlikely to be feasible. The arrangement of the letters in the square may be random, or it may be defined by a keyword. Figure 11.1 depicts the square based on the keyword PERSONALITY. Provided the sender and receiver can both remember the keyword, and the simple rules given below, the encryption and decryption procedures are straightforward. P N Y G U

E A B H V

R L C K W

S I D M X

O T F Q Z

Figure 11.1 The Playfair square with the keyword PERSONALITY In the encryption process spaces are ignored, and doubled letters are separated by inserting a dummy letter, such as X or Z. The plaintext is then split into digrams, and each digram xy is encrypted according to the following rules. (Minor variations are possible.) Case 1 Suppose x and y are in different rows and different columns. Then xy → ab where x, y, a, b are the corners of a rectangle, and x, a are in the same row. Case 2 Suppose x, y are in the same row. Then xy → uv, where u and v are the letters immediately to the right of x and y respectively. (If one of x and y is at the end of the row, the letter at the beginning of the row is used.) Case 3 Suppose x, y are in the same column. Then xy → uv, where u and v are the letters immediately below x and y respectively. (If one of x and y is at the bottom of the column, the letter at the top of the column is used.) For example, MA → HI BW → CV YF → BY SD → IM .

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The Playfair system is a symmetric key system. Both the encryption key and the decryption key are permutations of the set of 600 digrams; Alice uses a Playfair square and the rules given above to construct the key for encryption, and Bob uses the same square, but slightly different rules (Exercise 11.7) to construct the key for decryption. The two keys are inverse permutations.

Example 11.4 Suppose you receive the following ciphertext, knowing that the Playfair system has been used and the keyword is PERSONALITY. What is the plaintext? AQLAGCPZQTOLVAMLIYHSISEIHP Solution

The ciphertext broken up into a sequence of digrams is AQ LA GC PZ QT OL VA ML IY HS IS EI HP

.

Using the rules in reverse, and the Playfair square shown in Figure 11.1, the corresponding sequence of digrams is TH AN KY OU FO RT HE KI ND ME SX SA GE

.

Remembering the conventions about spaces and doubled-letters, it is clear that the plaintext is THANK YOU FOR THE KIND MESSAGE

.

As might be expected, it is possible to attack and break the Playfair system by frequency analysis. The attack uses the frequency table for digrams, and follows the same lines as the method using the frequency table for single symbols (Section 10.5). However, this procedure obviously requires more data than single-symbol analysis, and it takes longer to complete. Thus the system can provide an acceptable level of security in certain circumstances.

EXERCISES 11.5. Construct the Playfair square with keyword WHYSCRAMBLING, and use it to encrypt the plaintext THERE WILL BE A MEETING OF THE GROUP TOMORROW

.

11.6. Decrypt the following message, which has been encrypted by the Playfair rules with the same keyword as in the previous exercise. GMANBYZIIFCXWDHQNKENHLDUGKWRFMFUPGMH 11.7. Write down explicitly the rules for decryption corresponding to the rules for encryption given in the text above.

11.4 Mathematical algorithms in cryptography

185

11.4 Mathematical algorithms in cryptography Up to this point we have used mathematics mainly for the purpose of explaining how certain cryptosystems worked. In the twentieth century it gradually became clear that mathematics could be employed in a more fundamental way. We have already used the basic idea when we faced the problem of constructing codes with good error-correction properties. Thus, in Chapter 8 we allowed the symbols 0 and 1 in the binary alphabet B to have algebraic properties. We introduced the notation F2 for the field whose elements are 0 and 1 and F2 n for the vector space of n-tuples over F2 . The use of these algebraic constructions greatly extended the range of methods that could be employed. In general, the symbols in a message can be represented by the elements of an algebraic structure in many ways. The simplest way to represent a set of n objects by an algebraic structure is to use the integers mod n, denoted by Zn . For example, the 27-symbol alphabet A could be represented by 0, 1, 2, . . . , 26, regarded as elements of Z27 . We shall assume that the reader is familiar with fact that the integers mod n can be added and multiplied in such a way that they satisfy the usual rules of arithmetic. Technically, we say that Zn is a ring. In fact, it is often convenient to use an alphabet with a prime number of symbols, because when p is a prime number the integers mod p form a field. This means that every non-zero element has a multiplicative inverse. To emphasize its special nature we shall denote this field by Fp . Thus if we extend the alphabet A by allowing messages to contain commas and full-stops, in addition to the usual 27 symbols, then there are 29 symbols, a prime number. We can represent the space ⊔ by 0, the letters A, B, . . ., Z by 1, 2, . . . , 26, and the comma and full-stop by 27 and 28, all considered as elements of the field F29 . For example, if the message is I⊔CAME,⊔I⊔SAW,⊔I⊔CONQUERED. then we replace it by 9 0 3 1 13 5 27 0 9 0 13 1 23 27 0 9 0 3 15 14 17 21 5 18 5 4 28

.

The point is that, although arithmetic had no meaning when the symbols were ‘just’ symbols, we can now perform all the operations of elementary arithmetic, including division by a non-zero quantity. Furthermore, there are good algorithms for performing these operations, based on the methods we learned in elementary school. This possibility vastly increases the range of encryption methods that are available. A further possibility arises when we recall the standard technique of splitting the stream of symbols into blocks of an appropriate size m, so that each block becomes an m-vector over F29 . Now, not only can we ‘do arithmetic’ on the individual symbols, we can use linear algebra to manipulate the vectors.

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Example 11.5 Represent the message I⊔CAME,⊔I⊔SAW,⊔I⊔CONQUERED. as a string of 2vectors over F29 , and find the string of 2-vectors that results when the matrix   3 4 K= 2 3 is applied. How could the resulting string be transformed back into the original one? Solution

The message is represented by [9 0]′ [3 1]′ [13 5]′ · · ·

.

The dashes indicate that we shall regard the blocks as column vectors. Since      3 4 9 27 = , 2 3 0 18 and so on, this string is transformed into [27 18]′ [13 9]′ [1 12]′ · · ·

.

Given the second string, we can recover the first one by applying the inverse matrix, which in this case is     3 −4 3 25 K −1 = (3 × 3 − 4 × 2)−1 = . −2 3 27 3 The example illustrates the basic idea behind an early attempt to use abstract mathematical techniques in cryptography. In two papers published around 1930, L.S. Hill suggested that messages in the form given above could be encrypted by applying a linear transformation – that is, by multiplying the vectors by a matrix. Generally, in Hill’s system the key is an invertible m × m matrix K and the encryption function is EK (x) = Kx, where x is an m-vector. If y = Kx, then K −1 y = x. Hence the decryption function corresponding to EK is given by DL (y) = Ly, where L = K −1 . Hill’s sytem is another example of a symmetric key system, since K −1 can be computed from K using the standard methods of matrix algebra.

11.5 Methods of attack

187

EXERCISES 11.8. Represent the message AN⊔APPLE⊔A⊔DAY⊔KEEPS⊔DOCTOR⊔AWAY. as a string of elements of F29 2 . 11.9. Continuing from the previous exercise, calculate the first four blocks of the string that results when the plaintext is encrypted using the key   5 3 K= . 3 2

11.5 Methods of attack How might Hill’s system be attacked? An attack by exhaustive search would require checking all the m × m invertible matrices over F29 . An m × m matrix has m2 components, and if each component is an element of F29 , the number of 2 possibilities is 29m . Almost all these matrices are invertible, and consequently m can be chosen so that an attack by exhaustive search is not feasible. If exhaustive search is not feasible, the standard approach to a symmetric key cryptosystem is a known ciphertext (or ciphertext-only) attack. Eve obtains a piece of ciphertext c and tries to use this information in order to find the keys for encryption k and decryption ℓ. If these keys are found, Eve can not only decrypt the known ciphertext, using the rule Dℓ (c) = m, but also any other ciphertexts that are sent using the same keys. The attack by frequency analysis is a known ciphertext attack. But in the Hill system this attack is prevented by the fact that the letters are thoroughly scrambled. Any given sequence of letters will be encrypted in many different ways, depending on its position within a block and the other letters that occur in that block (see Exercise 11.11). However, a significant weakness of Hill’s system is the possibility of another form of attack: it may be possible for Eve to obtain some pieces of plaintext m and the corresponding ciphertexts c. Cryptographers recognise two kinds of attack based on this information. • In a known plaintext attack, Eve has obtained some pairs (mi , ci ), where each mi is plaintext and ci is the corresponding ciphertext. • In a chosen plaintext attack, Eve has obtained the ciphertexts ci corresponding to a number of specific plaintexts mi that she has chosen.

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In practice, the details will depend on the circumstances.

Example 11.6 Suppose Eve intercepts some ciphertext which (she believes) contains a report of an ambush, using the Hill system with m = 2. She suspects that the ciphertext [15 15]′ [25 15]′ [17 3]′ represents AMBUSH. How can she test her hypothesis and find the key? Solution

In 2-vectors over F29 the word AMBUSH is represented by

[1 13]′ [2 21]′ [19 8]′ .   a b Denote the key by K = . Eve’s hypothesis is that the sequence c d ′ ′ ′ [1 13] [2 21] [19 8] is encrypted as [15 15]′ [25 15]′ [17 3]′ . If this is true, the blocks AM and BU are encrypted as           15 a b 1 25 a b 2 = , = . 15 c d 13 15 c d 21 Consequently 

    15 25 a b 1 2 = , 15 15 c d 13 21     −1 a b 15 25 1 2 . = c d 15 15 13 21 Provided Eve can do some elementary algebra in F29 , she can calculate that     a b 2 1 K= = . c d 5 3 Eve can test her hypothesis first by checking that, with this K, the third block SH is encrypted as she suspects:      2 1 19 17 = . 5 3 8 3 Since this works, she can then apply K −1 =



3 −1 −5 2



to the rest of the ciphertext. If a meaningful plaintext results, the problem is solved. If not, Eve must try a different piece of ciphertext.

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189

EXERCISES 11.10. Eve routinely intercepts messages encrypted using the Hill system with blocks of size 2, and a key that is changed each day. The first message every day is a weather forecast that begins FORECAST⊔FOR ... . If today’s message begins with the ciphertext [13 7]′ [20 9]′ , what is today’s key? [Hint: in order to invert a matrix with determinant Δ ∈ F29 you must find the inverse of Δ in F29 .] 11.11. Suppose the Hill system is being used with blocks of size 5, and the key is the following matrix over F29 : 0 ⎜1 ⎜ K = ⎜ ⎜1 ⎝1 1 ⎛

1 0 1 1 1

1 1 0 1 1

1 1 1 0 1

⎞ 1 1⎟ ⎟ 1⎟ ⎟. 1⎠ 0

Show that the blocks AMPLE, DAMES, CRAMP, DREAM are encrypted in such a way that a different pair of elements of F29 appears in the positions corresponding to AM in each case. 11.12. Suppose Alice and Bob communicate using the Hill system with m = 3. What information must Eve obtain in order to make a successful known plaintext attack? If Eve wishes to use a chosen plaintext attack, suggest a good choice for the plaintexts. 11.13. Consider a cryptosystem in which the symbols are represented by elements of a field Fp and each symbol is encrypted by applying an encryption function of the form Eα,β (x) = αx + β

(α, β ∈ Fp , α = 0).

The key is the pair (α, β). Write down a suitable decryption function and explain why this is a symmetric key system. Explain how the system could be broken by a known plaintext attack. If it were possible to use chosen plaintext, what choice(s) would be be simplest?

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Further reading for Chapter 11 At this point the reader is advised to refer again to the historical accounts of cryptography listed at the end of Chapter 1. These books contain a great deal of valuable information on traditional cryptosystems, such as the poly-alphabetic ones, and the methods that have been used to attack them. The method of coincidences that is now used to break a Vig`enere cipher is largely the work of William Friedmann [11.1]. Hill’s system was described by him in two papers in 1929 and 1931 [11.2], [11.3]. 11.1 W.F. Friedman. The Index of Coincidence and its Applications in Cryptology. Dept. of Ciphers Publ. 22, Illinois, 1922. 11.2 L.S. Hill. Cryptography in an algebraic alphabet. Amer. Math. Monthly 36 (1929) 306-311. 11.3 L.S. Hill. Concerning certain linear transformation apparatus of cryptography. Amer. Math. Monthly 38 (1931) 135-154.

12 Cryptography in theory and practice

12.1 Encryption in terms of a channel The continued failure to construct an unbreakable cryptosystem led to attempts to analyse the process of encryption in mathematical terms. Shannon proposed a framework based upon the idea that converting plaintext into ciphertext can be regarded as transmission through a channel. Let us assume that there is a finite set M of plaintext messages that might be sent. Denote by pm the probability that the message is m: in other words, there is a probability distribution p on M. Note the implicit assumption that pm is not zero: messages that will never be sent are not included in M. We shall regard (M, p) as a memoryless source, which is the input to a ‘channel’, as follows. Let the set of possible keys be K, and let Ek be the encryption function for the key k ∈ K. Suppose that the probability that Ek is used is rk , so that we have a probability distribution r on the set K. It is assumed that the choice of k is independent of the message, m. The output of the system is the ciphertext c = Ek (m), hence the probability that c occurs is  qc = Pr(ciphertext is c) = pm rk , where the sum is over the set of pairs (m, k) such that Ek (m) = c. So we have a probability distribution q on the set C, and we can think of (C, q) as the output of a channel (Figure 12.1).

N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 12, 

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m Probability pm

Ek (m) = c where k has probability rk

c Probability qc

Figure 12.1 Encryption as transmission through a channel

If we regard p and q as row vectors in the usual way (see Chapter 5), the transformation effected by the encryption process is defined by a channel matrix Γ such that q = pΓ . Comparison with the formula for qc given above shows that we can define Γ as follows:  rk , Γmc = Pr(c | m) = k

where the sum is taken over all keys k such that Ek (m) = c. Note that Γ depends on the distribution r, just as (for example) the channel matrix for the extended BSC depends upon the bit-error probability e. In reality, the sets M, C, and K will be very large, but here is a small example for the purposes of illustration.

Example 12.1 Let M = C be the vector space F2 2 of ordered pairs xy, and take K = {1, 2} with r1 = r, r2 = 1 − r. Suppose the encryption functions are E1 (xy) = xy + 01,

E2 (xy) = xy + 10.

If the input distribution p on F2 2 is given by 00 01 10 11 0.1 0.2 0.3 0.4 what is the output distribution q? Solution Noting that in this example there is at most one k for each pair (m, c), and arranging the rows and columns in the order 00, 01, 10, 11, the

12.1 Encryption in terms of a channel

193

channel matrix Γ is 0 ⎜ r ⎜ ⎝1−r 0 ⎛

r 0 0 1−r

1−r 0 0 r

⎞ 0 1−r⎟ ⎟. r ⎠ 0

Hence the output distribution q = pΓ is 00 0.3 − 0.1r

01 0.4 − 0.3r

10 0.1 + 0.3r

11 0.2 + 0.1r

.

The basic method of attacking a cryptosystem is the known ciphertext attack (Section 11.5), in which Eve obtains a piece of ciphertext c and attempts to find the key k. Thus we are led to consider the uncertainty about k given c, or in the terms used in Chapter 5, the conditional entropy H(r | q). In this context it has a special name.

Definition 12.2 (Key equivocation) Given probability distributions p on M and r on K, the key equivocation of the system represented by Γ is H(r | q), where q = pΓ . The following lemma expresses the key equivocation in slightly simpler terms.

Lemma 12.3 In the notation used above, the key equivocation is given by H(r | q) = H(r) + H(p) − H(q).

Proof By definition, H(r | q) = H(j) − H(q), where j is the distribution of the pair (k, c). The entropy of j is the same as the entropy of the distribution of (k, m), since k and m determine c. Since the distributions of k and m are independent, this is equal to H(r) + H(p). Hence H(r | q) = H(j) − H(q) = H(r) + H(p) − H(q).

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As a first application of the channel formalism we consider the case where M is a set of meaningful plaintext messages in a natural language. Although our assumptions may appear rather optimistic, the results do provide a useful insight. We consider a system in which plaintext and ciphertext messages are strings in an alphabet X, and each plaintext is encrypted as a ciphertext of the same length, say n. In other words, M = C = X n . Provided that n is not too small (n = 10 will usually suffice), the following assumptions about the distributions p and q on X n are reasonable. 1. The entropy per symbol of the distribution on plaintexts, H(p)/n, is approximately U log2 |X|, where U is the uncertainty of the language (Definition 10.1). 2. All ciphertexts of length n are equally probable, so the entropy H(q) is log2 |X n | = n log2 |X|.

In this situation the key equivocation is given approximately by

H(r | q) ≈ H(r) + n(U − 1) log2 |X| = H(r) − nR log2 |X|, where R is the redundancy of the language (Definition 10.3). As n increases, the right-hand side decreases, and there is a value n0 at which it becomes zero. The value n0 is called the unicity point. It can be thought of as the length of a piece of ciphertext c that is sufficient to ensure there is no uncertainty: that is, there is only one message-key pair that could produce c. Using the estimate given above, we have n0 ≈

H(r) . R log2 |X|

It is possible to obtain numerical estimates for n0 by inserting some (rather speculative) values for the quantities on the right-hand side, but the significant conclusion is simply that a unicity point exists.

EXERCISES 12.1. Evaluate the key equivocation in Example 12.1 when r = 0.5. 12.2. Repeat the previous exercise with r = 0.51. Is it true that the key equivocation is greatest when r = 0.5? 12.3. Given the ciphertext c, let K(c) denote the set of keys k for which there is a plaintext m such that Ek (m) = c. If |K(c)| > 1 then all but one of the keys in K(c) is said to be spurious. In the Caesar system

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195

spurious keys can exist for ciphertexts of length 5 – see Exercise 10.12. Construct an example of a ciphertext of length 3 for which a spurious key exists. 12.4. Show that, in the Caesar system, there is a spurious key for a ciphertext of length 4, one of the plaintexts being the place-name ADEN.

12.2 Perfect secrecy It is reasonable to say that perfect secrecy occurs when the uncertainty about a piece of plaintext is not altered if the corresponding ciphertext is known. In the channel formalism, this translates into the fact that the uncertainty associated with the input source (M, p) should be the same as its uncertainty when the output source (C, q) is known. The latter quantity is the conditional entropy of p with respect to q, and as in Section 5.3 we denote it by H(Γ ; p) = H(p | q)

when q = pΓ.

Definition 12.4 (Perfect secrecy) Suppose that a cryptosystem is defined by a set of keys K and encryption functions Ek : M → C for k ∈ K. Suppose also that a probability distribution r on K is given. Then the system is said to have perfect secrecy if H(p) = H(Γ ; p) for all distributions p on M, where Γ is the channel matrix for the distribution r.

Theorem 12.5 A cryptosystem has perfect secrecy if and only if, for all probability distributions p on the set of plaintexts M, the channel matrix Γ for the distribution r on the set of keys K satisfies Γmc = qc

for all m ∈ M and c ∈ C,

where q is the corresponding distribution on the ciphertext space C.

Proof According to Theorem 5.11, H(p) = H(Γ ; p) if and only if p and q = pΓ are independent. This means that the probability tmc that the plaintext m and the

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ciphertext c occur is equal to pm qc . But we also have the equation tmc = Pr(c | m) pm = Γmc pm , so tmc = pm qc if and only if Γmc = qc , given our assumption that no messages with zero probability are included in M. The theorem suggests that perfect secrecy is very rare. The entries Γmc of the channel matrix are fixed, whereas the probabilities qc vary with the input distribution p. Hence the equality Γmc = qc can only hold in very special circumstances. An example will be given in the next section, but first here is a simple condition that follows directly from the theorem.

Corollary 12.6 If a cryptosystem has perfect secrecy then the number of keys is at least equal to the number of messages: |K| ≥ |M|.

Proof Fix c ∈ C such that qc > 0. Then the condition Γmc = qc implies that, for each m, the sum of rk taken over all keys k such that Ek (m) = c is not zero. In particular, for each m there is such a k. Since Ek is an injection, it follows that the keys for different m’s must be different. Hence there are at least as many keys as messages.

EXERCISES 12.5. Show that the system described in Example 12.1 does not have perfect secrecy, for any value of r. 12.6. Let M = C be the vector space F2 2 , and suppose the key space K is also F2 2 , with each key being equally probable. Suppose the encryption function for the key αβ ∈ F2 2 is Eαβ (xy) = xy + αβ. Show that this system has perfect secrecy. [This is a special case of the general result proved in the next section.]

12.3 The one-time pad

197

12.3 The one-time pad Perfect secrecy is a very restrictive condition, and examples are rare. But there is one important example, known as the one-time pad. In this system the plaintext and ciphertext sets M and C are both sets of n-tuples in a given alphabet, such as the english alphabet A. Since the theory works for any alphabet, for the purposes of exposition we shall use the binary alphabet F2 . We take M = C = F2 n , the vector space of strings of length n over F2 . The set of keys K is also F2 n , so Corollary 12.6 is satisfied. For each k ∈ K the encryption function Ek is m → m + k

(m ∈ M).

The corresponding decryption function is the same function, c → c+ k, because (m + k) + k = m. Clearly this is a symmetric key system. The idea behind the system is that the plaintext is expressed as a string m of n bits, and is encrypted by adding to it an arbitrary key string k of the same length. The next theorem says that the system has perfect secrecy if the key string is chosen uniformly at random. In other words, each new message is encrypted by using a new key, all keys being equally probable. That is the reason for the name one-time pad.

Theorem 12.7 The cryptosystem described above, with the probability distribution on the keys defined by 1 rk = n (k ∈ K), 2 has perfect secrecy.

Proof For any m ∈ M and c ∈ C there is a unique k such that Ek (m) = c, specifically k = m + c. It follows that the entries of the ‘channel matrix’ Γ are given by Γmc = rm+c =

1 2n

for all m, c.

Therefore the output distribution q is defined in terms of the input distribution p by  1 1  pm = n . qc = pm Γmc = n 2 2 m m

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Since qc = Γmc for all m and c, and for all p, it follows from Theorem 11.7 that the system has perfect secrecy.

EXERCISES 12.7. Describe briefly how the one-time pad system could be applied to messages in the 27-symbol english alphabet A. Illustrate your answer by decrypting the message RQOQXMMBDHK, given that the key is MATHEMATICS. 12.8. Suppose that a user of the one-time pad makes a mistake, and sends two different messages m1 and m2 using the same key. If Eve intercepts the ciphertexts c1 and c2 , what information does this provide? 12.9. Continuing from the previous exercise, suppose Eve suspects that somewhere in m1 a certain bit-string occurs – it might be the string representing TUESDAY for example. How can Eve test this hypothesis?

12.4 Iterative methods Throughout the twentieth century advances in the mathematical foundations of cryptography went hand-in-hand with technological developments. By using machines to scramble the plaintext very efficiently, it was hoped that it would be possible to achieve any desired level of security. Unfortunately, that hope was not realized. The capacity to devise complicated mechanisms for encryption and decryption always seemed to be matched by the capacity to devise mechanisms that can attack and break the resulting systems. The outcome was the conclusion that symmetric key cryptosystems can only provide security in a relative sense. The more complicated the procedures, the more difficult it is for an attack to be successful. But, at the same time, greater complication makes the system more difficult to use. Furthermore, systems can only be guaranteed secure against known methods of attack. On the technological side, the most significant development has been the availability of electronic computers. These are ideal for calculations that involve iteration, and we shall describe a very useful procedure that takes advantage of this fact. Suppose we are given a piece of text X, expressed as a string of n bits, and a key k, a string of s bits. Let F be a function that assigns to each such X and

12.4 Iterative methods

199

k another string of n bits, Y = F (k, X). In other words we have a function F : F2 s × F2 n → F2 n . For reasons of security, F should not be a ‘nice’ function, such as a linear function. When s = 2 and n = 3 we might use the function defined for a key k = αβ and X = x1 x2 x3 by the rule y1 = αx1 x2 + β,

y2 = x2 + βx3 ,

y3 = (α + β)x1 x3 .

For a given key, this function is not linear, nor is it a bijection.

Definition 12.8 (Feistel iteration) The Feistel iteration associated with F and the initial values X0 , X1 ∈ F2 n , is defined as follows. Let k = (k1 , k2 , . . . , kr ) be a sequence of keys, each of which is an element of of F2 s , and define Xi+1 = Xi−1 + F (ki , Xi )

(i = 1, 2, . . . , r),

where + denotes the operation of bitwise addition in the vector space F2 n .

Figure 12.2 The ith round of a Feistel iteration

In cryptography each step of the iteration is called a ‘round’. In the ith round the key ki is used to transform the pair Xi−1 Xi into Xi Xi+1 (Figure 12.2). Encryption is done by expressing all messages as sequences of strings of

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bits of length 2n. If the plaintext contains a 2n-string m, it is split into two equal parts, m = X0 X1 , and the n-strings X0 and X1 are taken as the initial values for the Feistel iteration. The corresponding ciphertext is the outcome after r rounds: c = Xr Xr+1 . Thus the encryption function for a key-sequence k is given by Ek (m) = c,

where m = X0 X1 , c = Xr Xr+1 .

Before we discuss the practical details, here is a simple example.

Example 12.9 Take s = n = 3, and define Y = F (k, X) for a key k = αβγ by the rule y1 y2 y3 = F (αβγ, x1 x2 x3 ), y1 = αx1 + x2 x3 ,

y2 = βx2 + x1 x3 ,

where y3 = γx3 + x1 x2 .

If the key-sequence is k = (100, 101, 001, 111) and m = 101 110, calculate c = Ek (m). Solution

The calculation can be tabulated as follows. i

ki

Xi

0 1 2 3 4

− 100 101 001 111

101 110 000 110 001

F (ki , Xi ) Xi+1 − 101 000 001 001

110 000 110 001 111

Thus c = 001 111. The Feistel iteration provides unlimited scope for scrambling the data, since the parameters n, s, r can be as large as we please. Nevertheless it is trivially a symmetric key system!

Theorem 12.10 Suppose the message m = X0 X1 is encrypted using the Feistel system with a function F and the key-sequence k = (k1 , k2 , . . . , kr ), so that c = Ek (m) = Xr Xr+1 . Then the Feistel system with the same function F and key sequence k∗ = (kr , kr−1 , . . . , k1 ), when applied to c′ = Xr+1 Xr , yields m′ = X1 X0 .

12.5 Encryption standards

201

Proof Since we are working in F2 n , the equation for the Feistel iteration can be rewritten as Xi−1 = Xi+1 + F (ki , Xi ). In this form the iteration proceeds in reverse: starting with i = r the equation gives Xr−1 in terms of Xr+1 and Xr , and so on. Hence if the original key sequence is used in reverse, the same procedure can be used for decryption. We have already noted that the function F should not have ‘nice’ properties. For instance, if it is a linear function, the Feistel system will be vulnerable to a known plaintext attack (Exercise 12.12). As always, the security of the system will also depend on there being enough keys to guard against an attack by exhaustive search. All the keys k1 , k2 , . . . , kr could be chosen independently, but in practice it is usual to ‘extract’ them by choosing specific sets of s bits from a master key K. For example, if 12 keys of length 32 are needed, Alice and Bob could agree on a master key of length 120, and take k1 as bits 1 to 32, k2 as bits 9 to 40, k3 as bits 17 to 48, and so on.

EXERCISES 12.10. Take s = 2, n = 3, and define Y = F (k, X) for a key k = αβ by the rule y1 y2 y3 = F (αβ, x1 x2 x3 ) where y1 = β(x1 + x2 ),

y2 = (α + β)x2 ,

y3 = α + x2 x3 .

If the key-sequence is k = (11, 10, 10, 01) and m = 011 001, calculate c = Ek (m). 12.11. Check your answer to the previous exercise by applying the relevant decryption function to c. 12.12. Suppose that, for any given key k, the function X → F (k, X) is a linear function. Show that the entire Feistel iteration based on F is a linear function.

12.5 Encryption standards In the 1970s the increasing use of cryptographic procedures in commerce and industry led to the suggestion that an ‘encryption standard’ should be established. In general, standards are used to communicate essential information

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to everyone, such as how to make nuts and bolts that will fit together. In cryptography, the aim is more subtle, because Alice and Bob must be able to communicate privately, without letting Eve know exactly how they are doing it. Our discussion thus far has indicated that any cryptosystem must depend for its security on the secrecy of the key. Consequently, a system that purports to provide a standard of security must be prepared to answer two basic questions. • Is the attack by exhaustive search feasible? • Is there an attack that is better than exhaustive search?

In 1977 the first Data Encryption Standard, usually known as DES, was established. A central component is what is known as an S-box. In general terms, this is an array (matrix) S with rows corresponding to elements of F2 M , columns corresponding to elements of F2 N , and entries in F2 R . In DES there are 8 S-boxes, with parameters M = 2, N = 4, and R = 4. One of them is shown in Figure 12.3. For clarity, the rows are labelled 0 − 3 as binary numbers, 0 = 00, 1 = 01, 2 = 10, 3 = 11. The column labels and the entries of S are denoted by 0 − 15 in the same way: thus 5 = 0101, 12 = 1100, and so on. 0 0 1 2 3

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

12 1 10 15 10 15 4 2 9 14 15 5 4 3 2 12

9 7 2 9

2 6 8 0 13 3 4 14 7 5 11 12 9 5 6 1 13 14 0 11 3 8 8 12 3 7 0 4 10 1 13 11 6 5 15 10 11 14 1 7 6 0 8 13

Figure 12.3 One of the S-boxes used in DES Each S-box in DES determines a function fS : F2 6 → F2 4 , defined by the rule fS (x1 x2 x3 x4 x5 x6 ) = S(x1 x6 , x2 x3 x4 x5 ). For example, for the S-box given above, fS (111001) = S(11, 1100) = S(3, 12) = 6 = 0110. The DES encryption function operates on blocks of 64 bits, using a key with (essentially) 56 bits. It is obtained by permuting the bits and combining the 8 functions fS defined by the S-boxes in a complicated way, involving a Feistel iteration with 16 rounds. The details can be found in the references cited at the end of the chapter.

12.6 The key distribution problem

203

The S-boxes were constructed so that they satisfied a number of criteria, designed to make the system safe against known methods of attack. Two of these criteria were • C1: each row of S is a permutation of F2 4 ;

• C2: if x, y ∈ F2 6 are such that the Hamming distance d(x, y) = 1, and x′ = fS (x), y ′ = fS (y), then d(x′ , y ′ ) ≥ 2.

The proposal for DES led to much controversy. The method of constructing the S-boxes was not revealed, a fact that some people regarded as suspicious. More seriously, the parameters chosen were thought to be too small, mainly because the master key had effectively only 56 bits. This meant that an attack by exhaustive search was not out of the question. In fact, by the 1990s it was clear that DES could not answer either of the basic questions satisfactorily. Consequently it was replaced by the Advanced Encryption Standard or AES. In this system the master key has 128 bits, and it is currently not feasible to carry out an attack by exhaustive search. The system is not based directly on a Feistel iteration, but similar principles are applied. There is only one S-box, with M = N = 4 and R = 8 and it is defined by an explicit mathematical formula. So, if there are any ‘hidden’ features, everyone has an equal chance of discovering them. It will be interesting to see how long AES can survive.

EXERCISES 12.13. Let S be the S-box shown in Figure 12.3. Determine the values of fS (x) and fS (y) when x = 101100, and y = 101101, and verify that condition C2 holds in this case. 12.14. Let S be an S-box with M = N = 2 and R = 3, and let fS (x1 x2 x3 x4 ) = S(x1 x4 , x2 x3 ). Show that the appropriate form of condition C2 implies that the four entries in any row or column of S must be a code in F2 3 with minimum distance 2. Hence construct a suitable S.

12.6 The key distribution problem In a symmetric key cryptosystem, Alice and Bob use similar keys. Metaphorically speaking, Alice puts the message in a box which she locks using the key

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k. When Bob receives the box he unlocks it, using a key ℓ that is closely related to k. Writing ℓ = k ′ to emphasize this relationship, the procedure is illustrated in Figure 12.4.

Ek

Dk′

? Figure 12.4 The symmetric key procedure A major problem is that Alice and Bob must begin by communicating with each other in order to agree on the key, and this communication cannot therefore be protected by the encryption process. This fundamental difficulty is known as the key distribution problem. There are several ways in which the key distribution problem might be overcome. If enormous resources are available, the one-time pad system could be used. In other words, before each message is transmitted Alice conveys a new key to Bob by some highly secure means, such as a courier protected by armed guards. Clearly, this is impractical for most purposes. A more practical approach is to consider alternatives to the symmetric key procedure. Question: Is it possible to design a procedure in which Alice’s key and Bob’s key are independent, so that neither of them needs to know the other’s key? At first sight, the following procedure appears to work. 1. Alice sends Bob a message encrypted using her key a. 2. Bob further encrypts using his key b and returns the message to Alice. 3. Alice decrypts using a′ and sends the message to Bob. 4. Bob decrypts using b′ . This double-locking procedure is illustrated in Figure 12.5. Unfortunately, our use of the metaphor about locking and unlocking boxes has concealed a serious difficulty. Denote the respective encryption and decryption functions by Ea , Da′ , Eb , Db′ , where Da′ is the left-inverse of Ea and Db′ is the left-inverse of Eb . Then the process applied to a message m is represented by the transformation m → Db′ Da′ Eb Ea (m).

If it happens that Da′ Eb = Eb Da′ , this transformation will reduce to the identity, since in that case Db′ Da′ Eb Ea (m) = Db′ Eb Da′ Ea (m) = m. Here we use

12.6 The key distribution problem

205

ALICE

BOB

Ea

Eb

Da Db

Figure 12.5 The double-locking procedure the fact that, by definition, both Da′ Ea and Db′ Eb′ are the identity. In other words, the double-locking procedure will work if Alice’s operations commute with Bob’s. It appears to work in the diagram because in practical terms the operations of turning keys in separate locks do commute. Unfortunately, mathematical operations are not necessarily so well-behaved. For example, suppose the encryption and decryption keys are permutations of 26 letters, as in Section 10.5. Then it is easy to construct permutations that do not commute (see Exercise 12.15). Note that if Alice and Bob agree to restrict themselves to a set of permutations that do commute, then they can use the double-locking procedure. For example, they could use the Caesar system, in which each key is a permutation of the form αk , where α is the permutation written in cycle notation as α = (ABCD · · · XYZ). In that case we have a set of 26 commuting keys αk (1 ≤ k ≤ 26). But this number is far too small to be useful in practice, and there remains the problem of how Alice and Bob can agree on which key to use. In the next chapter we shall describe a procedure that avoids the key distribution problem in a entirely different way.

EXERCISES 12.15. In Section 10.5 we described how permutations can be defined in terms of keywords. Alice has chosen the keyword DEMOCRAT and Bob has chosen REPUBLICAN. Write down the corresponding permutations δ and ρ in cycle notation, and show that the double-locking system fails in this case.

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12. Cryptography in theory and practice

Further reading for Chapter 12 Perhaps the first advance with implications for modern cryptography was the introduction of the one-time pad. The original version was a practical system suggested by Vernam in 1917, and later patented. Soon afterwards Mauborgne observed that if the key is chosen uniformly at random then the system will have good security. In 1949 Shannon applied his seminal ideas to the science of cryptography, and gave a mathematical proof of this fact [12.4]. The idea was well known by the time of World War II, and a version of it was employed in the Lorenz machine used by the German high command. In 1941 an incompetent operator sent two slightly different messages with the same key (see Exercises 12.8 and 12.9), and this contributed significantly to the breaking of the system by Tutte and others at Bletchley Park, the British Cryptographic HQ [12.7]. In the 1960s the one-time pad (with suitably draconian safeguards) was used to transmit messages between Moscow and Washington. There are several interesting websites devoted to this topic. Comprehensive accounts of current practice in cryptography, with details of DES, AES, and many topics not mentioned here, can be found in the books by Menezes et al. [12.3], Stinson [12.5], and Trappe and Washington [12.6]. The official specifications for DES [12.2] and AES [12.1] are also worth reading. 12.1 Advanced Encryption Standard (AES). Federal Information Processing Standard (FIPS), Publication 197 (2001). 12.2 Data Encryption Standard (DES). Federal Information Processing Standard (FIPS), Publication 46 (1977). 12.3 A.J. Menezes, P.C. van Oorschott, S.A. Vanstone. Handbook of Applied Cryptography. CRC Press (1996). 12.4 C. Shannon. Communication theory of secrecy systems. Bell Syst. Tech. J. 28 (1949) 657-715. 12.5 D.R. Stinson. Cryptography, Theory and Practice. Chapman and Hall (third edition, 2006). 12.6 W. Trappe and L.C. Washington. Introduction to Cryptography and Coding Theory. Prentice-Hall (2003). 12.7 W.T. Tutte. Fish and I. In: Coding Theory and Cryptology (D. Joyner ed.). Springer, New York (2000).

13 The RSA cryptosystem

13.1 A new approach to cryptography In the previous chapter we noted that symmetric key cryptosystems are limited by the problem of distributing keys. A radical new approach was developed in the 1970s, known as public key cryptography. The fundamental idea is that a typical user (Bob) has two keys, a public key and a private key. The public key is used by Alice and others to encrypt messages that they wish to send to Bob, and the private key is used by Bob to decrypt these messages. The security of the system depends on ensuring that Bob’s private key cannot be found easily, even though everyone knows his public key. A practical method of implementing this idea was discovered by Rivest, Shamir, and Adleman in 1977 and is known as the RSA cryptosystem. There are now many other systems of public key cryptography, but RSA is still widely used, and it is worth studying because it illustrates the basic principles simply and elegantly. In the RSA system the plaintexts are sequences of integers mod n, where it is reasonable to think of n as a number with at least 300 decimal digits. Any simple rule can be used to convert the ‘raw’ message to plaintext. For instance, if the message is written in the 27-symbol alphabet A, we could begin by representing each symbol as a 5-bit binary number: ⊔ → 00000,

A → 00001,

B → 00010,

...,

Z → 11010.

N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 13, 

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13. The RSA cryptosystem

Then we could form blocks of (for example) 180 symbols, making strings of 900 binary digits, which represent numbers with about 300 decimal digits. These numbers can then be treated as integers mod n, provided n is large enough. In practice, a system based on the ASCII or Unicode standards would be used: the point is that the plaintext can be constructed, and converted back into its ‘raw’ form, using publicly available information. We have already observed on several occasions that the set of integers mod n, with the operations of addition and multiplication, is a ring. We denote it by Zn . In general, not every non-zero element of Zn has a multiplicative inverse. We assume that the reader is familiar with the fact that a non-zero element x ∈ Zn has a multiplicative inverse x−1 ∈ Zn if and only if gcd(x, n) = 1. In Section 13.3 we shall describe a good method of calculating x−1 .

Definition 13.1 (φ function) For any positive integer n, the number of integers x in the range 1 ≤ x ≤ n such that gcd(x, n) = 1 is denoted by φ(n). It follows from the result stated above that φ(n) is also the number of invertible elements of Zn . For example, taking n = 14 the values of x such that gcd(x, 14) = 1 are 1, 3, 5, 9, 11, 13 and so φ(14) = 6. In this case the inverses are easily found by inspection: thus 3−1 = 5 since 3 × 5 = 1 (mod 14), and so on. If n is a prime number, say n = p, then the p − 1 numbers x = 1, 2, . . . , p − 1 all satisfy gcd(x, p) = 1, and so φ(p) = p− 1. We shall need the following simple extension of this rule.

Lemma 13.2 If n = pq, where p and q are primes, then φ(n) = (p − 1)(q − 1).

Proof The integers x such that gcd(x, n) = 1 are the multiples of p and the multiples of q. In the range 1 ≤ x ≤ n these are: p q

2p 3p 2q 3q

. . . (q − 1)p . . . (p − 1)q

qp pq

(there are q of these); (there are p of these).

Since qp = pq = n, the total number is p + q − 1, so we have φ(n) = n − (p + q − 1) = pq − p − q + 1 = (p − 1)(q − 1).

13.2 Outline of the RSA system

209

EXERCISES 13.1. Make a list of those elements of Z36 that have multiplicative inverses, and find the inverse of 5 mod 36. 13.2. Evaluate φ(257) and φ(253). 13.3. Given that x has the inverse y mod n, find the inverse of n − x mod n. Deduce that φ(n) is an even number, except when n = 1 and n = 2.

13.2 Outline of the RSA system In the RSA system there are a number of users, including, as always, Alice and Bob. Each user, say Bob, has an encryption function and a decryption function, constructed according to the following rules. • Choose two prime numbers p, q and calculate n = pq,

φ = (p − 1)(q − 1).

• Choose e such that gcd(e, φ) = 1, and calculate d = e−1 (mod φ). The encryption and decryption functions are defined as follows: En,e (m) = me Dn,d (c) = cd

(m ∈ Zn ), (c ∈ Zn ).

The system works in the following way. Starting with p and q, Bob uses the rules given above to construct, in turn, the numbers n, φ, e, and d. He makes his public key (n, e) available to everyone, but keeps his private key d secret. When Alice wishes to send Bob a message, she expresses it in the form of a sequence of integers m mod n, calculates c = En,e (m), and sends c. Bob then uses his private key to compute Dn,d (c). (Note that n is not private, but it is needed in the construction of Dn,d .) The following example uses very small numbers and is presented for the purposes of illustration only. The calculations can be checked by hand (although some hard work is needed in part (iii)). The information that Bob must keep private is in bold type.

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13. The RSA cryptosystem

Example 13.3 Suppose Bob has chosen p = 47, q = 59. (i) Find n and φ, and show that e = 157 is a valid choice for his public key. (ii) Verify that his private key is d = 17. (iii) How does Alice send Bob a message represented by a sequence of integers mod n, containing (for example) the integer m = 5? How does Bob decrypt it? Solution

(i) We have n = 47 × 59 = 2773,

φ = 46 × 58 = 2668.

The choice e = 157 is valid provided that gcd(2668, 157) = 1. This condition holds because 157 is a prime and does not divide 2668. (ii) We can check this by calculating de = 17 × 157 = 2669 = 1 (mod 2668). (iii) If Alice wishes to send Bob a message, she looks up his public key (n, e) = (2773, 157), and converts her message into a sequence of integers mod n. She encrypts using the encryption function En,e (m) = me , and sends the ciphertext to Bob. In this case c = 5157 = 1044 (mod 2773). When Bob receives the ciphertext c = 1044 he applies his (private) decryption function Dn,d (c) = cd , obtaining m′ = 104417 = 5

(mod 2773).

Since m′ = m, the original message is recovered. In reality, the numbers must be much larger, and Bob would require the help of a computer algebra system, such as MAPLE, to set up as a user. To find the two primes p and q, he could use the MAPLE command nextprime(x), which returns the next prime number greater than x. For instance, the command >p:= nextprime(13∧151); q:= nextprime(19∧132); will result in the output p= q=

160489 · · · 624417 · · ·

··· ···

· · · 323547 · · · 576463

where there are over 150 digits in each number. The significant fact is that the numbers p and q are found almost instantaneously, so this computation can

13.2 Outline of the RSA system

211

be said to be ‘easy’. The reason is far from obvious: it is a consequence of the existence of good algorithms for primality testing, the details of which can be found in the references given at the end of the chapter. It is rather less surprising that, with the aid of MAPLE, Bob can easily calculate n = pq: >n:= p*q; n = 100212 · · ·

where n has over 300 digits.

···

···

· · · 874261

In order to justify the usefulness of RSA as a practical cryptosystem, it remains to provide satisfactory answers to three questions. • Feasibility Why is it easy to carry out the calculations involved, such as finding d = e−1 (mod φ), and computing the values of me and cd (mod n)? • Correctness Why is the function Dn,d the left-inverse of En,e ? That is, why is it true that Dn,d (En,e (m)) = m

for all m ∈ Zn ?

• Confidentiality Why cannot another user (Eve) discover the plaintext m, given that she knows the public key (n, e) and the ciphertext c = En,e (m)? These questions will be answered in the next three sections, on the assumption that the number n = pq is ‘large’. The following exercises illustrate the fact that the system is vulnerable if n small.

EXERCISES 13.4. George and Tony used the RSA cryptosystem. George advertised his public key n = 77, e = 43. What were the primes p and q, and what was his private key? Tony sent him a message m ∈ Zn , and Vladimir intercepted the ciphertext c = 5. What was m? 13.5. Tony advertised the public key n = 3599, e = 31. Unfortunately, he has become confused as to whether his PIN (private key) is 3301, 3031, or 3013. Which is it? 13.6. A naive user of RSA has announced the public key n = 2903239, e = 5. Eve has worked out that n = 1237 × 2347. Verify that the public key is valid and explain why the private key is d = 2319725. [Calculators are not required.]

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13.3 Feasibility of RSA We shall now explain why the calculations involved in RSA can be done easily. This depends on the existence of two algorithms, both of which are used to reduce complicated problems to basic arithmetic. The Euclidean algorithm is essentially a method for calculating the greatest common divisor gcd(a, b) of two integers a and b (we can assume a < b). It depends on the fact that if b = qa + r then gcd(a, b) = gcd(r, a). We can therefore replace (a, b) by (a′ , b′ ) = (r, a), and repeat the process. Eventally we obtain a pair (a∗ , b∗ ) in which a∗ is a divisor of b∗ .

Example 13.4 Find the greatest common divisor of 654 and 2406. Solution

The calculation proceeds as follows: 2406 654 444 210 24 18

= 3 × 654 = 1 × 444 = 2 × 210 = 8 × 24 = 1 × 18 = 3 × 6.

+444 +210 +24 +18 +6

Consequently 6 is the greatest number that divides 18, 24, 210, 444, 654, 2406, and in particular gcd(654, 2406) = 6. If gcd(a, b) = 1, then a has a multiplicative inverse mod b, and the Euclidean algorithm can be used to calculate it. This is done by reversing the calculation, so that 1 is expressed in the form λa + μb, where λ and μ are integers. The equation 1 = λa + μb can be written in the form λa = 1 − μb, that is, λa = 1 (mod b). Thus λ is the inverse of a (mod b).

Example 13.5 Find the inverse of 24 mod 31. Solution

First we check that gcd(24, 31) = 1: 31 24 7 3

= 1 × 24 +7 = 3 × 7 +3 = 2 × 3 +1 = 3 × 1.

13.3 Feasibility of RSA

213

Now, starting with the last-but-one equation and working backwards, we have 1 =7−2×3 = 7 − 2 × (24 − (3 × 7)) = 7 × 7 − 2 × 24 = 7 × (31 − 1 × 24) − 2 × 24 = −9 × 24 + 7 × 31. Hence (−9) × 24 = 1 (mod 31), so 24−1 = −9 = 22.

Example 13.6 In Example 13.3 Bob set up as a user of RSA by choosing p = 47, q = 59, so that n = 2773, φ = 2668. Use the Euclidean algorithm to show that e = 157 is a valid choice for his public key, and explain how he can calculate his private key d = e−1 (mod φ). Solution The Euclidean algorithm shows that gcd(e, φ) = 1: 2668 = 16 × 157 +156 157 = 1 × 156 + 1 156 = 156 × 1 . d = e−1 is found by working backwards: 1 = 157 − 1 × 156 = 157 − 1 × (2668 − (16 × 157)) = (−1) × 2668 + 17 × 157. Thus 17 × 157 = 1 (mod 2668), so d = 17 is Bob’s private key. In the real world the numbers are huge, and a computer must be used. It can perform the calculations very quickly. For example, in MAPLE the command >d:= e∧ (-1) mod phi; will return the value of d if e has an inverse mod φ, or an error message if not. The second useful algorithm is repeated-squaring, which is used to calculate the power bk , where b and k are given positive integers. In RSA the processes of encryption and decryption both involve calculation of this kind, since the plaintext m and the ciphertext c are related by the equations c = me , m = cd . Naively, it seems that in order to compute the power bk we must use k − 1 multiplications, since bk = b × b × · · · × b. However the repeated squaring algorithm provides a much better way.

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13. The RSA cryptosystem

Example 13.7 Suppose the number b is given. How many multiplications are needed to compute b16 and how many more multiplications are needed to compute b23 ? Solution

The calculation of b16 requires only four multiplications: b2 = b × b,

b4 = b2 × b2 ,

b8 = b4 × b4 ,

b16 = b8 × b8 .

Given this calculation, we can use the fact that 23 = 16 + 4 + 2 + 1, so that only three more multiplications are needed to find b23 : b23 = b16 × b4 × b2 × b. Consequently b23 can be found with only 4 + 3 = 7 multiplications in all. The method used in the example can be used to calculate any power bk , as follows. The first stage is to calculate, in turn, i

b 2 , b 4 , b 8 , . . . , b2 , where 2i is the largest power of 2 not exceeding k. This requires i multiplications. At the second stage k is expressed as a sum of a subset of the numbers 1, 2, 4, . . . , 2i , and the corresponding powers of b multiplied to give bk . This stage requires at most i multiplications, and consequently the total number is at most 2i. Since 2i ≤ k, the bound 2i is approximately 2 log2 k. This number is proportional to the number of digits required to represent k (rather than k itself), because writing k in the usual decimal representation requires log10 k digits. Since log10 k = log2 k / log2 10, it follows that for a number with ℓ decimal digits, the number of multiplications is at most (2 log2 10) × ℓ ≈ 6.64 ℓ. For example, for a number with 150 digits, about 103 multiplications are needed. By comparison the naive method would require 10150 . In the application to RSA there is the further complication that multiplication must be carried out modulo an integer n. After each multiplication, the remainder on division by n must be found. In MAPLE this operation can be implemented by the single command >b &∧ k (mod n);

.

For future reference it is worth noting that, although these calculations are feasible, they cannot be said to be trivial. We shall return to this topic in Section 15.5.

13.4 Correctness of RSA

215

EXERCISES 13.7. Use the Euclidean algorithm to show that gcd(15, 68) = 1. Hence find the inverse of 15 mod 68. 13.8. Given a positive integer b, explain how to calculate b55 with only 9 multiplications, using the repeated-squaring algorithm. 13.9. Approximately how many multiplications are needed to calculate b3578674567 by the repeated-squaring algorithm? 13.10. A naive user of RSA has announced the public key n = 187, e = 23. Find the factors of n and hence (i) verify that the public key is valid and (ii) determine the private key. If the user receives the integer 6 (mod 187) as a piece of ciphertext, what number m represents the plaintext? What function was used by the sender to encrypt m, and how many multiplications are required to calculate this function? 13.11. Calculate 21000 mod 47. [MAPLE can be used, but it is possible to do the calculation ‘by hand’.] 13.12. The repeated-squaring algorithm does not necessarily use the least possible number of multiplications. For example, there is a method of calculating b55 using only 8 multiplications, rather than the 9 used in Exercise 13.8. Find this method. [See also Section 15.5.]

13.4 Correctness of RSA The next task is to explain why RSA works – that is, why the function Dn,d is the left-inverse of En,e . According to the definitions, the condition Dn,d (En,e (m)) = m reduces to (me )d = m (mod n), so we have to prove that this holds when d is the inverse of e mod φ(n).

Lemma 13.8 If gcd(x, n) = 1, so that x has an inverse in Zn , then xφ(n) = 1 in Zn .

Proof Suppose the invertible elements of Zn are b1 , b2 , . . . , bf , where we write f =

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13. The RSA cryptosystem

φ(n) for short. Given that gcd(x, n) = 1, x is invertible, and each xbi is also −1 invertible, since the inverse of xbi is b−1 . Thus the sets {b1 , b2 , . . . , bf } i x and {xb1 , xb2 , . . . , xbf } have the same members. Since multiplication in Zn is commutative the product of the elements is the same, whatever the order, and so b1 b2 · · · bf = (xb1 )(xb2 ) · · · (xbf ) = xf (b1 b2 · · · bf ) in Zn . Hence xf = 1, and since f = φ(n) the result follows.

In the following proof we assume that the plaintext element m ∈ Zn is such that gcd(m, n) = 1. In practice n = pq, where p and q are primes with at least 150 digits, so the probability that an element of Zn does not satisfy this condition is (p + q − 1)/pq (Lemma 13.2). This is of the order of 10−150 , so the probability is infinitesimally small. (In fact the result does hold when gcd(m, n) = 1, but a slightly different proof is needed – see Exercise 13.15.)

Theorem 13.9 If the RSA encryption and decryption functions are En,e and Dn,d , and gcd(m, n) = 1, then Dn,d (En,e (m)) = m. That is, med = m (mod n).

Proof In the RSA construction, starting with p, q, Bob calculates n = pq and φ = (p − 1)(q − 1), so φ = φ(n), by Lemma 13.2. Bob then chooses e so that gcd(e, φ) = 1, so that e has an inverse d mod φ. Explicitly ed = 1 + φα,

for some integer α.

It follows from Lemma 13.8 that med = m1+φα = m × (mφ )α = m

(mod n),

as claimed.

EXERCISES 13.13. If n = 15 and e = 3, what is d? Verify explicitly that in this case Dn,d (En,e (m)) = m for all m ∈ Zn , including those values of m for which gcd(m, n) = 1.

13.5 Confidentiality of RSA

217

13.14. Show that the statement mφ(n) = 1 (mod n) does not hold for all m in the range 1 ≤ m < n when n = 15. 13.15. Let n be the product of primes p and q. Let m be such that gcd(m, n) = 1 and 1 ≤ m < pq, and let K be a multiple of φ(n). Show that mK+1 = m (mod p),

mK+1 = m (mod q).

Deduce that the RSA encryption-decryption process works even when gcd(m, n) = 1.

13.5 Confidentiality of RSA It remains to explain why Eve cannot discover the plaintext m, given that she knows the public key (n, e) and may obtain the ciphertext c = En,e (m). By definition m = Dn,d (c) = cd (mod n). In order to calculate m, it would be enough for Eve to know the private key d, which requires knowledge of φ, since d is the inverse of e mod φ. This in turn requires knowledge of the primes p and q. But Eve knows only n, not the factorization n = pq. Although there is a good algorithm for multiplying p and q (long multiplication, as taught in elementary arithmetic) no one has yet found a good algorithm for ‘unmultiplying’: that is, for finding p and q when n is given. (It is worth noting that, as a consequence of the ‘Fundamental Theorem of Arithmetic’, p and q are unique.) For example, if Eve has access to MAPLE, she could attempt to find the factors p and q by using the command >ifactor(n); which will work if n is small (see Exercise 13.16). But if n has 300 decimal digits the algorithms currently available (2008) will not respond to Eve’s command, even if she is prepared to wait for a lifetime. This suggests that the problem is ‘hard’. Thus, for the time being, Eve cannot expect to break RSA by factorizing n. The possibility of a successful attack by a different method remains open. In summary, the ‘secret components’ for each user are the pair (p, q), and the values of φ and d. In fact, if Eve obtains any one of these three quantities, she can calculate the others, and the security of the system for that user will be destroyed. For example, if Eve knows φ, then not only can she calculate d, she can also calculate p and q, as in the following example.

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13. The RSA cryptosystem

Example 13.10 Bob is using RSA with public key n = 28199, e = 137. Eve has discovered that Bob’s value of φ is 27864. Find the primes p, q such that n = pq. Solution

We have pq = 28199,

(p − 1)(q − 1) = 27864,

so that p + q = 336. Given the sum and product of p and q, it follows that they are the roots of the quadratic equation x2 − 336x + 28199 = 0. Solving this equation by factorizing the quadratic expression is no easier than factorizing 28199. But fortunately we know a formula that reduces the problem to the calculation of a square root:   1 p, q = 336 ± 3362 − 4 × 28199 . 2 There is a good algorithm for calculating square roots, although it is not really needed in this example, since 3362 − 4 × 28199 = 112896 − 112796 = 100. So the roots are (336 ± 10)/2, that is p = 173, q = 163. We have given plausible reasons why RSA can be implemented as a working cryptosystem. It is worth repeating that the most significant property, confidentiality, is not a mathematically proven fact. It relies on the presumed difficulty of factoring integers, and if there is a significant advance in that direction, the system might be compromised.

EXERCISES 13.16. The Department of Bureaucracy has advertised that it will use the RSA cryptosystem with the public key n = 173113, e = 7. Using the following MAPLE output, show that the Department’s choice of public key is valid, and find the private key. >ifactor(173113); (331)(523) 13.17. Bob is using RSA with public key n = 4189, e = 97. Eve has discovered that Bob’s value of φ is 4060. Find the primes p, q such that n = pq and Bob’s private key d.

13.5 Confidentiality of RSA

219

13.18. Chang is using RSA with public key n = 247 064 529 085 306 223 003 563, e = 145 268 762 498 836 504 194 121. Deng has discovered that the private key d = 987654321 will successfully decrypt messages addressed to Chang. Explain how, without factorizing n, Deng can also obtain φ and the primes p, q used by Chang to construct the keys. 13.19. A user of the RSA system has mistakenly chosen a public key (n, e) with n prime. Show that messages sent to this user can be decrypted easily. 13.20. Although the general problem of factorizing a number with (say) 500 digits is still thought to be intractable, there are some numbers with 500 digits for which some factors can be found relatively easily. For example, consider the number N = 12345678901234567890 · · · 1234567890, where there are 50 blocks of the digits 1234567890. Can you find factors of N ? Can you find the prime factors of N ? Can you find the prime factors of N + 1?

Further reading for Chapter 13 The idea of public-key cryptography was first suggested by Diffie and Hellman [13.4] in 1976. About a year later Rivest, Shamir, and Adleman proposed a practical system, and it was published by Martin Gardner in the Scientific American [13.5]. In fact a system very similar to RSA had been suggested a few years earlier by three mathematicians, Cocks, Ellis, and Williamson, working for the British cryptographic service (GCHQ). But this fact remained secret and was not made public until 1997: for the details see Singh [1.3, 279-292]. The discovery of RSA stimulated a great deal of research into the computational aspects of number theory. A standard reference is the book by Crandall and Pomerance [13.3]. From the RSA perspective there are two major questions. The first is the factoring problem, the difficulty of which is the basis for the belief that the system is secure. As an illustration, Gardner [13.5] challenged readers of the Scientific American to factorize a number with 129 digits. This number was not factorized until 1994, by the combined efforts of over 600

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people [13.2]. The factoring problem is still believed to be ‘hard’, but there is as yet no mathematical proof of this statement. The second problem is primality testing, which is needed to construct the primes p and q. For this problem probabilistic methods that work well in practice, such as the Solovay-Strassen algorithm [13.9] and the Miller-Rabin algorithm [13.7, 13.8], have been known since the 1970s. More recently Agarwal, Kayal, and Saxena [13.1] succeeded in constructing a deterministic algorithm that is technically ‘good’, but the probabilistic methods are more effective in practice. There are numerous attacks on RSA that use special features of the numbers involved. Also, the traditional ingenuity of codebreakers has also resulted in suggestions for attacks based on quite different considerations – analogous to the attack on a substitution system by frequency analysis. Such methods are known as side-channel attacks. One of the first was discovered by Kocher [13.6], then a student at Stanford, who noticed that the inner workings of the repeatedsquaring algorithm could be revealed by recording how much time was needed to do the calculations. 13.1 M. Agarwal, N. Kayal, N. Saxena. PRIMES is in P. Annals of Math. 160 (2004) 781-793. 13.2 D. Atkins, M.Graff, A. Lenstra, P. Leyland. The magic words are squeamish ossifrage. Advances in Cryptology - ASIACRYPT ’94 (1995) 703-722. 13.3 R. Crandall and C. Pomerance. Prime Numbers: A Computational Perspective. Springer-Telos (2000). 13.4 W. Diffie and M. Hellman. New directions in cryptography. IEEE Trans. in Information Theory 22 (1976) 644-654. 13.5 M. Gardner. A new kind of cipher that would take millions of years to break. Scientific American 237 (August 1977) 120-124. 13.6 P. Kocher. Timing attacks on implementations of Diffie-Hellman, RSA, DSS and other systems. Advances in Cryptology - CRYPTO ’96 (1996) 104-113. 13.7 G.L. Miller. Riemann’s hypothesis and tests for primality. Journal of Computer and Systems Science 13 (1976) 300-317. 13.8 M.O. Rabin. Probabilistic algorithms for testing primality. Journal of Number Theory 12 (1980) 128-138. 13.9 R. Solovay and V. Strassen. A fast Monte Carlo test for primality. SIAM Journal on Computing 6 (1977) 84-85.

14 Cryptography and calculation

14.1 The scope of cryptography Modern cryptography is not just about sending secret messages. It covers many aspects of security, including authentication, integrity, and non-repudiation. • Authentication Bob must be sure that a message that purports to come from Alice really does come from her. • Integrity Eve should not be able to alter a message from Alice to Bob. • Non-repudiation Alice should not be able to claim that she was not responsible for a message that she has sent. In this chapter we shall consider how the ‘public key’ approach can be applied to these topics. The general framework for a public key cryptosystem is a simple abstraction from the RSA system discussed in the previous chapter. There are sets M, C, and K, of plaintext messages, ciphertext messages and keys, respectively, and a typical user (Alice) has two keys: a public key a and a private key a′ . These keys determine an encryption function Ea and its left-inverse, a decryption function Da′ : Da′ (Ea (m)) = m for all m ∈ M. The security of the system depends on the fact that, given a, it is hard to find a′ . Unfortunately, the precise definition of the words ‘easy’ and ‘hard’ is rather complicated. The branch of theoretical computer science known as complexity N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 14, 

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theory is concerned with making mathematically sound statements about ‘easy’ and ‘hard’ computations, but the theory is still being developed. For expository purposes it is enough to rely on practical experience of computer algebra systems such as MAPLE. In Chapter 13 we used this approach to back up the claim that RSA is a practicable system, based on the fact that multiplying is easy, but unmultiplying is hard. In Section 14.4 we shall describe another operation that is easy to do, but hard to undo. As we shall see, it forms the basis of many security procedures used in modern cryptography.

14.2 Hashing A simple device that is often useful in cryptography is known as hashing. The idea is that a message, which may be of any length, is reduced to a ‘message digest’, which is a string of bits with a fixed length. We shall explain how this process can be used to authenticate a message, and to guarantee its integrity – with some provisos.

Definition 14.1 (Hash function) Let n be fixed positive integer (in practice, the value n = 160 is often used). A hash function for a set of messages M is a function h : M → F2 n . A simple application of this construction is that when Alice wishes to send Bob a message m, she actually sends the pair (m, h(m)). This pair may or may not be encrypted by Alice and decrypted by Bob in the usual way; in either case Bob should obtain a pair (x, y) with x = m and y = h(m). If Bob gets a pair (x, y) such that h(x) = y, he knows that something is wrong. Note that it is quite possible that the ‘false’ message x is meaningful: it may have been sent by Eve, with the intention of causing trouble. Since messages can have any length, we can assume that the size of M is greater than 2n . This means that h cannot be an injective function: there will surely be two different messages m and m′ such that h(m) = h(m′ ). If h is constructed simplistically, then it will be easy to find examples.

Example 14.2 Suppose a hash function h is constructed as follows. The message m is expressed as a string of bits, and split into blocks of length n (using some 0’s to fill in the last block, if necessary). Then h(m) defined to be the sum (in F2 n ) of the

14.2 Hashing

223

blocks. Given m ∈ M explain how to construct m′ ∈ M such that m′ = m and h(m′ ) = h(m). Solution by adding block and produce a h(m).

There are many ways in which this can be achieved. If m′ is obtained a specific string s of n bits to two blocks in m, then s + s is the zero h(m′ ) = h(m). This procedure can be extended in obvious ways to message that is apparently very different from m, but has the same

In cryptography, a pair m = m′ such that h(m) = h(m′ ) is known as a collision. Clearly, in order to guarantee the authenticity and integrity of messages it is important that collisions are hard to find, even though they are bound to exist. A hash function that has these properties is said to be collision-resistant. Many such functions have been proposed, and some of them have been widely used. However, it is worth noting that collisions have been constructed in several of the most popular hash functions. The hashing procedure can be strengthened significantly if Alice and Bob agree to use a family of hash functions {hk }, where the key k ∈ K is secret.

Definition 14.3 (Message authentication code) A message authentication code (MAC) is a family of hash functions hk (k ∈ K) such that (i) given m ∈ M and k ∈ K, it is easy to compute hk (m); (ii) given a set of pairs (mi , hk (mi )) (1 ≤ i ≤ r), but not k, it is hard to find hk (m) for any m that is not one of the mi . The conditions are designed for the situation in which Eve’s goal is to construct a false message m with a valid hash hk (m). If Eve is lucky she may be able to intercept any number of messages mi and their hashes hk (mi ). Thus it is important that this information will not enable her construct hk (m). We shall return to the topics of authentication, integrity, and non-repudiation in Section 14.7.

EXERCISES 14.1. Let h : M → F2 n be a hash function, and define μy = |{x | h(x) = y}| (y ∈ F2 n ),

σ = |{{x1 , x2 } | h(x1 ) = h(x2 )}|.

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Note that σ counts the collisions as the number of unordered pairs. If M = |M|, show that  y

μy = M,

σ=

1 2 M . μy − 2 y 2

14.2. Deduce from the results of the previous exercise that the number of M (M − N ), where N = 2n . collisions is at least 2N 14.3. Show that if {hk } (k ∈ K) is a MAC then each hk must be collisionresistant. 14.4. Suppose that M is a vector space and h is a linear transformation M → F2 n . Show that h does not satisfy condition (ii) of Definition 14.3.

14.3 Calculations in the field Fp We have already made use of the fact that when p is prime, the set of integers mod p, with the relevant arithmetical operations, is a field, denoted by Fp . Specifically, this means that every element of Fp except 0 has a multiplicative inverse, and so the set of non-zero elements of Fp forms a group with the operation of multiplication. This group will be denoted by F× p , and its members by 1, 2, . . . , p − 1 in the usual way. It is important to remember that there is a distinction between the positive integer n and the corresponding element of F× p , which we also denote by n, although a notation like [n]p would be be more correct. The two objects behave differently under the respective arithmetical operations. For example, the integer 20 is such that 20 × 20 = 400, whereas when 20 is taken as an element of F× 23 (say), we have 20 × 20 = 9. The distinction is even more obvious when we consider multiplicative inverses. In order to define the multiplicative inverse 1 . of the integer 20 we have to use a different kind of number, the fraction 20 × On the other hand, the multiplicative inverse of 20 as an element of F23 is 15, since 20 × 15 = 1 (mod 23). From a practical point of view, the significant point is that some, but not all, calculations in F× p can be done by adapting the familiar methods that we use for ordinary integers. For example, to find 20 × 20 in F× 23 we multiply as

14.3 Calculations in the field Fp

225

if 20 was an ordinary integer, and then reduce mod 23 (that is, we find the remainder when 400 is divided by 23): 20 × 20 = 400 = 17 × 23 + 9 = 9 (mod 23). Similarly, in order to find 20−1 in F× 23 we can also use ordinary arithmetic, by applying the Euclidean algorithm (Section 13.3). This gives the result 20−1 = 15, which can be checked by another familiar calculation: 20 × 15 = 1 + 13 × 23. However, there are some calculations in F× p for which (as yet) no one has found a way to employ the familiar algorithms of arithmetic. The most important example arises from the following definition.

Definition 14.4 (Primitive root) A primitive root of a prime p is an element r of the group F× p such that the p − 1 powers of r r, r2 , r3 , . . . , rp−1 are all distinct, and therefore comprise all the elements of F× p. A famous result (conjectured by Euler, and proved by Legendre and Gauss) guarantees that for every prime p there is at least one primitive root r.

Example 14.5 Find all the primitive roots of 11. Solution Clearly 1 is not a primitive root, so the first candidate is 2. A simple calculation produces the table 2 22 2 4

23 8

24 5

25 26 10 9

27 7

28 3

29 6

210 1

.

Thus 2 is a primitive root. In general, a primitive root r must have the property that r10 is the smallest power of r which is equal to 1, and the table enables us to check this condition. For example, 3 is not a primitive root, since 3 = 28 ,

and so

35 = 240 = (210 )4 = 14 = 1.

Using similar arguments it is easy to show that 4, 5, 9, and 10 are not primitive roots, whereas 6, 7, and 8 are. Hence there are four primitive roots 2, 6, 7, 8. This is a particular case of the general result (Exercise 14.6) that the number of primitive roots of p is φ(p − 1).

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EXERCISES 14.5. Show that 3 is a primitive root of 17, and hence find all the primitive roots of 17. 14.6. Use your calculations from the previous exercise to find the inverse of every element of F× 17 . 14.7. Alice and Bob have agreed to express their messages as sequences of elements of F× p . They choose keys a, b (1 ≤ a, b ≤ p − 1), and their encryption functions are Ea (m) = ma ,

Eb (m) = mb

(m ∈ F× p ).

What conditions on a and b must be satisfied in order that there should exist suitable decryption functions Da′ and Db′ of a similar form? If these conditions are satisfied, explain how Alice and Bob can avoid the key distribution problem by using the double-locking procedure described in Section 12.6. 14.8. Let r be a primitive root of p. Prove that rk is also a primitive root if and only if gcd(k, p − 1) = 1. Deduce that the number of primitive roots of p is φ(p − 1).

14.4 The discrete logarithm Suppose a prime p and a primitive root r are given, and consider the ‘exponential’ function x → rx . Given x, we can compute the function value y = rx by using the repeated-squaring algorithm (Section 13.3). The obvious way to attack the reverse problem, given y, find x such that rx = y, is by ‘brute force’ – working out the powers of r in turn, until the correct value of x is found. Methods of solving the DLP that are better than brute force are known, but there is as yet no general method that will work for large numbers. The general problem is known as the Discrete Logarithm Problem or DLP. The name arises from the fact that (in real analysis) the logarithm function is the inverse of the exponential function. In our case we can define the discrete logarithm of y ∈ F× p , with respect to the primitive root r, to be the integer x such that rx = y

in F× p

and 1 ≤ x ≤ p − 1.

14.4 The discrete logarithm

227

The MAPLE command >mlog(y,r,p) attempts to find the logarithm of y to base r, in F× p . However, the current implementation is such that it will usually fail to produce an answer for large values of p. Intuitively, the difficulty arises because the behaviour of the exponential function x → rx on F× p is quite unlike the behaviour of the corresponding function on the field of real numbers. In the real case the function is continuous: given that 24 = 16 (for example), then we can infer that the real number x for which 2x = 17 is close to 4. By comparison, the behaviour of x → 2x on F× p is quite irregular, even for small values of p, such as p = 11 (Example 14.5).

Example 14.6 Show that 5 is a primitive root of 23, and find the logarithms to base 5 of 2 and 3 (mod 23). Solution In order to show that 5 is a primitive root it is sufficient to show that the order of 5 (the least value of x such that 5x = 1) is 22. It follows from elementary Group Theory that the order must be a divisor of 22, so we have only to check x = 2 and x = 11. Now 52 = 2

and 511 = (52 )5 × 5 = 25 × 5 = 9 × 5 = 45 = −1.

Since 52 = 1 and 511 = 1, the order of 5 is indeed 22. Since 52 = 2, it follows that log5 2 = 2. But this does not help us to find log5 3, so brute force is the simplest way: 52 = 2, 53 = 10, 54 = 4, . . . , 516 = 3, . . . . Hence log5 3 = 16. As an illustration, here is a simple application of the DLP in computer science, a method of storing passwords on a server. Suppose a set of users wish to login securely to a server. The administrator chooses a prime p and a primitive root r, and each user chooses a password that can be kept secure, either by memorizing it or recording it in a safe way. The password for each user (Alice) is converted by some automatic process into an element πA of F× p . It should be noted that if the ‘raw’ passwords are chosen too predictably – for example, if Alice chooses a real word such as WONDERLAND – then the system will be vulnerable to an attack based on searching a dictionary. The administrator then calculates σA = rπA (mod p) and stores the list of pairs (A, σA ) on the server. The values of p and r will also be kept on the server. Alice can authenticate her identity by sending her password, which is converted to the form πA , so that the server can check that rπA = σA (mod p).

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This system provides a degree of security, even if a hacker succeeds in obtaining all the information stored on the server, including p, r, and σA . Given this information it is still hard to determine Alice’s password πA , because that would entail solving the DLP, rx = σA (mod p).

EXERCISES 14.9. Verify that 2 is a primitive root of 19, and find the logarithms (to base 2) of 12 and 15 (mod 19). 14.10. An incompetent administrator has set up a password scheme based on p = 101 and the primitive root r = 2. Eve hacks into the server and discovers these values, and the value stored against Alice’s name, which is σA = 27. What is Alice’s password πA ? 14.11. Let logr x denote the logarithm of x to base r, where r is a primitive root for a given prime p. Prove that logr (ab) = logr a + logr b (mod p − 1).

14.5 The ElGamal cryptosystem In 1985 ElGamal showed that a public key cryptosystem can be based on the Discrete Log Problem. It is assumed that all messages are expressed as elements of F× p in a standard way (for long messages several elements may be needed), and a fixed primitive root r of p is known to all users. Each user, such as Bob, chooses a private key b′ ∈ N, and computes his public key b ∈ F× p by the rule ′ b = rb . An important feature of ElGamal’s system is that the encryption of a message involves a token, t ∈ N, which is chosen randomly each time the system is used. When Alice wishes to send Bob a message m she chooses t ∈ N, and using the publicly available values of r and b, she applies the encryption function Eb (m, t) = (rt , mbt ). Thus Bob receives ciphertext in two parts: the first part is the ‘leader’ ℓ = rt , and the other part is the encrypted message c = mbt .

14.5 The ElGamal cryptosystem

229

Lemma 14.7 The decryption function Db′ defined by ′

Db′ (ℓ, c) = c (ℓ−1 )b . is the left-inverse of the encryption function Eb defined above, for all t ∈ N.

Proof We have to check that Db′ (Eb (m, t)) = m for all m ∈ F× p and all t ∈ N: ′

Db′ (Eb (m, t)) = Db′ (rt , mbt ) = (mbt )((rt )b )−1 ′ = (mbt )((rb )t )−1 = (mbt )(bt )−1 = m.

Example 14.8 Suppose p = 1009, r = 102, and Bob has chosen the private key b′ = 237. (i) What is Bob’s public key? (ii) If Alice wishes to send the message m = 559 to Bob, and chooses the token t = 291, what ciphertext will Bob receive? (iii) How will Bob decrypt the ciphertext? Solution

(i) Bob’s public key is b = 102237 = 854.

(ii) If Alice uses Eb to send him the message m = 559 with the token t = 291, then Bob will receive the ciphertext (rt , mbt ) = (102291 , 559 × 854291 ) = (658, 23). (iii) On receiving (658, 23) Bob will apply his decryption function Db′ (ℓ, c) = ′ c (ℓb )−1 , obtaining 23 × (658237 )−1 = 23 × 778−1 = 23 × 463 = 559. Clearly, the security of the ElGamal system depends on the difficulty of the DLP. If Eve can solve the DLP to find x such rx = b (mod p) then Bob’s private key b′ is x, and Eve can apply the decryption function Db′ . Other forms of attack are also possible, and the parameters must be chosen with these in mind. For example, if Alice always uses the same value of t, then the system is vulnerable (see Exercise 14.14).

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EXERCISES 14.12. Suppose I am using an experimental version of the ElGamal cryptosystem, with p = 10007 and r = 101. I choose my private key to be k ′ = 345. What is my public key? 14.13. Continuing with the set-up described in the previous exercise, suppose another user wishes to send me the message x = 332, and chooses the token t = 678. What is the encrypted form (ℓ, c) of the message, as I receive it? Verify that my rule for decryption correctly recovers the original message. 14.14. Show that if Alice always uses the same value of t then the ElGamal system can be broken by a known plaintext attack, with one piece of plaintext m1 and the corresponding ciphertext (ℓ1 , c1 ).

14.6 The Diffie-Hellman key distribution system In Section 12.6 we noted that one of the problems of symmetric key cryptography is the distribution of the keys. If Alice and Bob wish to communicate, they must first agree on a key, and that agreement appears to require a separate form of communication. In 1976 Diffie and Hellman proposed a key-distribution system that avoids this problem. Suppose a set of users wish to communicate in pairs, using a symmetric key system. Each pair, such as Alice and Bob, must choose a key kAB in such a way that no other user has direct knowledge of it. The Diffie-Hellman system begins by assuming that a large prime p and a primitive root r have been selected, and they are public knowledge. Alice then chooses a private key a′ ∈ N, and computes ′

a = ra . Alice declares a to be her public key. Similarly, Bob chooses b′ ∈ N and declares ′ his public key b = rb . When Alice and Bob wish to communicate, they use the key ′ ′

kAB = ra b . Observe that Alice can easily calculate kAB using her own private key a′ and ′ Bob’s public key b, since kAB = ba . Similarly, Bob can calculate kAB using the ′ formula ab .

14.6 The Diffie-Hellman key distribution system

231

On the other hand, suppose Eve wishes to discover kAB . She knows the values a, b, and so it would be enough to solve either of the Discrete Log Problems rx = a,

rx = b,

which would give her a′ or b′ . But if the numbers are large enough, these problems are thought to be hard. (There might also be a quite different method of solving this particular problem, but as yet no one has found it.)

Example 14.9 Suppose Fiona, Georgina, and Henrietta have agreed to encrypt their text messages using the Diffie-Hellman system with p = 101 and r = 2. They have chosen the private keys f ′ = 13, g ′ = 21, h′ = 30, respectively. (i) What common information will be stored in the directory of each girl’s phone? (ii) What key will be used for messages between Georgina and Henrietta, and how do they obtain it? (iii) How could Fiona eavesdrop on messages between Georgina and Henrietta? Solution (i) Each directory will contain the values p = 101, r = 2, and the public keys f, g, h, computed by the rules f = 213 , g = 221 , h = 230 , as follows. f 12

g 89

h 17

(ii) The key for communication between Georgina and Henrietta is kGH = 221×30 = 17. ′

′

Georgina can calculate this as hg = 1721 and Henrietta as g h = 8930 . (iii) If Fiona wishes to discover kGH she could try to find either g ′ or h′ , by solving one of the equations 2x = 89, 2x = 17, or she could use some other (as yet unknown) method.

EXERCISES 14.15. Four people, A, B, C, D, have chosen to communicate using the Diffie-Hellman system, with p = 149 and r = 2. If A has chosen the private key 33, what is her public key?

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14.16. Continuing with the set-up described in the previous exercise, suppose the public keys of B, C, D are 46, 58, 123 respectively. What key should A use to communicate with B, and how does she obtain it? If A wishes to discover the key used for communication between C and D, what Discrete Log Problems might she try to solve?

14.7 Signature schemes In this section we return to the aspects of cryptography discussed at the beginning of this chapter, with particular emphasis on the topic of non-repudiation. Suppose Alice wishes to send the message m to Bob, who would like to have irrefutable evidence that the message really does come from Alice. Clearly, she must include in the message her ‘real’ name, so that Bob will be aware of the purported sender. But she can also add a signature, y = s(m), in a form that confirms her identity. Her definition of s(m) must remain private, otherwise anyone could forge her signature. In order to verify that a message m accompanied by a signature y is really signed by Alice, Bob must be able to decide whether or not s(m) = y without knowing precisely how Alice constructed the function s. Note that this procedure is quite different from the traditional use of hand-written signatures. In that case a signature is a fixed object, independent of the message, whereas here the relationship between the signature and the message is crucial. Also, it is useful for Alice to be able to use several different signatures. The context for the following definition is a communication system in which a typical user A has a public key a and a private key a′ .

Definition 14.10 (Signature scheme) A signature scheme for a set M of messages comprises a set Y of signatures and • for each user A, a set SA of signature functions s : M → Y, such that A can calculate s(m) for s ∈ SA and m ∈ M using the private key a′ ; • a verification algorithm that, given m ∈ M and y ∈ Y, enables any other user to check the truth of the statement s(m) = y for some s ∈ SA using only the public key a. We shall say that y is a valid signature for a message m from A if the statement is true.

14.7 Signature schemes

233

In order to see how the scheme works, we begin by explaining how, under certain conditions, a public-key cryptosystem can be adapted for use as a signature scheme. Suppose a cryptosystem is given and, for each user A, let SA consist of a single signature function s, which is actually the user’s decryption function Da′ . Since we require a signature function to have domain M and range Y, and Da′ is in fact a function from C to M, we must assume that M = C = Y here. Using Da′ as the signature function, Alice can send the signed message (m, Da′ (m)) = (m, y) to Bob in the usual way, using his (public) encryption function Eb to encrypt both parts. So Bob receives the ciphertext in the form (Eb (m), Eb (y)). He decrypts this ciphertext using his own decryption function Db′ , and recovers (m, y). The first part m is meaningful plaintext, but the second part y is apparently unintelligible. However Bob knows that the message purports to come from Alice, because the meaningful plaintext says so. He also knows Alice’s public key a, and his problem is how to check that y is a valid signature, using a, but not her private key a′ . Suppose Bob applies Ea to y: Ea (y) = Ea (Da′ (m)). If it happens that Ea (Da′ (m)) = m then Bob will have another copy of m. Since only Alice could have constructed Da′ (m), Bob is sure that Alice really is the sender of m. What is more, he can prove it, by producing both m and Da′ (m). However, the condition that Ea (Da′ (m)) = m for all m ∈ M does not hold in general. We do know that Da′ is the left-inverse of Ea , Da′ (Ea (m)) = m

for all m ∈ M,

because that is the basic property of a cryptosystem, but the condition required here involves the operators in reverse order. In other words, the condition is that the operators commute. To summarize, we have proved the following theorem.

Theorem 14.11 Suppose there exists a public-key cryptosystem with the properties that M = C and, for each user A, the functions Ea and Da′ commute. Then there is an associated signature scheme in which Y = M and • SA contains just one signature function Da′ ; • any user can check that y is a valid signature for the message m from A by making the test: is Ea (y) = m?

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Example 14.12 Show that the commutativity property holds in the RSA cryptosytem. Solution In RSA the encryption and decryption functions for A are given by En,e (x) = xe , Dn,d (x) = xd , where x ∈ Zn . Hence En,e (Dn,d (x)) = (xd )e = (xe )d = Dn,d (En,e (x)). This result shows that RSA can be used as a signature scheme. In practice, the signature is usually applied to a hash of the message, h(m) (represented as an element of Zn in some standard way), rather than to m itself. Alice’s signature is y = h(m)d (mod n), and Bob’s verification test is y e = h(m)

(mod n)?

On the other hand, the commutativity property does not hold for the ElGamal cryptosystem, so that system cannot be used without modification. ElGamal himself explained how to make the changes required. ElGamal’s signature scheme is defined in the following theorem. As with the cryptosystem, all users are assumed to know a prime p and a primitive root r, and the set of messages is taken to be M = F× p . The user A has a private key ′ ′ × a ∈ N and a public key a ∈ Fp , related by a = ra . There are many signature functions for A, rather than just one. Specifically, the set SA contains functions st that depend upon a suitably chosen parameter t ∈ N: SA = {st | 1 ≤ t ≤ p − 1 and gcd(t, p − 1) = 1}. In the following theorem it is useful to distinguish between an element of F× p and the integer that represents it. We adopt the temporary notation that |x| denotes the unique integer that represents x ∈ F× p and satisfies 1 ≤ |x| ≤ p − 1.

Theorem 14.13 (ElGamal signature scheme) Let t and u be integers such that 1 ≤ t, u ≤ p − 1 and tu = 1 (mod p − 1). Let Y = F× p × N and define the signature function st : M → Y by st (m) = (i, j) where i = rt , j = u(|m| − a′ |i|). Then (i, j) is a valid signature for the message m from A if and only if a|i| ij = r|m|

in F× p.

14.7 Signature schemes

235

Proof If (i, j) is a valid signature, then i = rt for some value of t. Thus ′

′

a|i| ij = ra |i| rtj = ra |i|+tj . Since tu = 1 (mod p − 1) we have a′ |i| + tj = a′ |i| + tu(|m| − a′ |i|) = |m| (mod p − 1), as claimed. Conversely, suppose m and (i, j) are such that a|i| ij = r|m| . Since i is in F× p we have i = rt for some t such that 1 ≤ t ≤ p − 1. Hence ′

′

r|m| = a|i| ij = ra |i| rtj = ra |i|+tj . Since r has order p − 1 this implies that |m| = a′ |i| + tj

(mod p − 1),

that is

j = u(|m| − a′ |i|).

EXERCISES 14.17. Suppose the ElGamal signature scheme is used with p = 23 and r = 5. Alice’s public key is a = 6. Is (2, 4) a valid signature for the message m = 6 from Alice? 14.18. Show that the number of valid signatures for a given message in the ElGamal scheme is φ(p − 1). In the previous exercise, what are the possible values of i in a valid signature (i, j), and why is i = 2 not one of them? 14.19. Suppose that, in the ElGamal system, Eve knows Alice’s public key a but not her private key a′ , and she has a fake message m that she wants Bob to think has been sent by Alice. There are two simple strategies that might allow her to construct a valid signature (i, j). (1) Choose i randomly and attempt to calculate the correct j. (2) Choose j randomly and attempt to calculate the correct i. Show that the first strategy is equivalent to solving a DLP. What equation must be solved in order to succeed with the second strategy? [It is not known how this problem is related to the DLP.]

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14.20. Alice and Bob have agreed to use the RSA signature scheme (as described above), and Alice’s public key is n = 187, e = 3. She has arranged with Bob that they will use a hash function h with values in F2 8 , regarded as the binary representation of numbers mod 187. (For example, 00101100 is regarded as 44.) Bob has received two messages purporting to come from Alice. For one the hash is 00001011, and the signature is 9, and for the other the hash is 00010101 and the signature is 98. Write down Bob’s verification test and decide whether either of the messages could be genuine.

Further reading for Chapter 14 The relationship between cryptography and complexity theory is explained thoroughly in the book by Talbot and Welsh [14.3]. The surveys of cryptography listed at the end of Chapter 12, especially those by Menezes et al. [12.3], Stinson [12.5], and Trappe and Washington [12.6], also contain a great deal of relevant material. The Diffie-Hellman key-distribution system based on the DLP was described in the original paper on public key cryptography [13.4]. Soon afterwards Pohlig and Hellman showed that solving the DLP on any cyclic group of order n could be reduced to solving the DLP on the cyclic groups of order f , where f is a prime factor of n. This implies that when applications that rely on the DLP on F× p are designed, the prime p should be chosen so that p − 1 has very few prime factors. Designers must also be aware of the attack by index calculus. This is a method specific to the group F× p that can produce results if p is chosen unwisely; it is the described in the cryptography texts mentioned above. ElGamal’s algebraic formulae for F× p were published in 1985 [14.1]. In the next chapter we shall explain how the same formulae can be applied more generally. 14.1 T. ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. on Information Theory 31 (1985) 469-472. 14.2 S.C. Pohlig and M.E. Hellman. An improved algorithm for computing logarithms over GF (p) and its cryptographic significance. IEEE Trans. on Information Theory 24 (1978) 106-110. 14.3 J. Talbot and D. Welsh. Complexity and Cryptography. Cambridge University Press (2006).

15 Elliptic curve cryptography

15.1 Calculations in finite groups In this chapter we continue to adopt the cryptographic perspective developed in the earlier chapters. This means that we stress the link between mathematical theory and practical calculation which, as we have seen, is fundamental in modern cryptography. The specific system that we consider is based on groups that arise in the theory of elliptic curves, a topic that fascinated mathematicians for over a century before it first found practical application in the 1980s. From the cryptographic perspective, it is worth stressing that a group consists of two things: a set of elements G, and an operation ∗ defined on pairs of elements (h, k) in G. When we say that a group (G, ∗) is ‘given’, we mean that we know how to represent h and k in a definite way, and how to calculate h ∗ k using this representation. Ultimately, it must be possible to reduce these calculations to operations on strings of bits, although it is usually convenient to use a more familiar notation, such as the standard decimal notation for integers. For example, suppose we are working in F× p , the group of nonzero elements of Fp , under multiplication. Then the elements of F× p are represented by natural numbers 1, 2, . . . , p − 1 in decimal notation, and h × k can be calculated by applying the familiar ‘long multiplication’ algorithm, followed by ‘long division’ to find the remainder when the result is divided by p.

N. L. Biggs, An Introduction to Information Communication and Cryptography, c Springer-Verlag London Limited 2008 DOI: 10.1007/978-1-84800-273-9 15, 

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Definition 15.1 (Cyclic group, generator) The group (G, ∗) is a cyclic group of order n, with generator g if |G| = n and there is an element g ∈ G such that the elements of the group are equal to g, g 2 , g 3 , . . . , g n , in some order. (It follows that the group is abelian, and g n is the identity element.) The powers of any element h ∈ G are defined recursively by the rule h1 = h,

hi = h ∗ hi−1 (i ≥ 2).

Although the definition suggests that i − 1 applications of the ∗ operation are needed in order to calculate hi , the repeated squaring algorithm (Section 13.3) allows hi to be calculated much more efficiently. Furthermore, if G has n elements, every h ∈ G is such that hn is equal to the identity element. Thus calculating the inverse of h can, if necessary, be done by using the rule h−1 = hn−1 . The fact that F× p is a cyclic group of order p − 1 is a consequence of the famous result that there is a primitive root r (a generator of F× p ) for every prime p. However, there are two problems. Finding a primitive root r for a given p is not trivial, and the correspondence between the powers of r and the elements of F× p is complicated. The latter problem is just the Discrete Logarithm Problem discussed in the previous chapter. As we shall see, both these problems occur more generally in elliptic curve cryptography. The following example illustrates the fact that, without some additional information, finding a generator for a cyclic group may require ‘brute force’ methods.

Example 15.2 Let ∗ denote multiplication of 2 × 2 matrices. Find g such that the following set of six matrices forms a cyclic group (G, ∗) with generator g. √       1 3 1 0 −1 0 2√ 2 1 0 1 0 −1 − 23 2 √ √ √   1  1  1 3  3  −2 − − 23 − 2 2 2 2 √ √ √ 3 3 1 − 21 − 23 − 21 2 2 2

Solution If we are unaware of the geometrical significance of the matrices, we must proceed by working out the orders of the matrices. The first two matrices have orders 1 and 2 respectively, and clearly they are not generators. However,

15.2 The general ElGamal cryptosystem

239

if we take the third matrix to be g, then it turns out that the six given matrices are equal to g, g 2 , g 3 , g 4 , g 5 , g 6 (but obviously not in the given order). Thus g is a generator. Suppose the problem of finding a generator g in a cyclic group G has been solved. Then we might hope to use the correspondence between the elements of G and the powers of g to simplify calculations in the group. Thus in order to calculate h ∗ k we can write h = g i , and k = g j , calculate i + j using ordinary addition, and set h ∗ k = g i+j . But this method assumes that we can find i and j, the ‘logarithms’ of h and k. That is precisely the Discrete Log Problem (DLP) for (G, ∗): given a generator g ∈ G and any h ∈ G, find i ∈ N such that g i = h. In some cases this problem may be easy (Exercise 15.3), but in general it is hard.

EXERCISES 15.1. Show that the operation of multiplying complex numbers makes the following set of numbers into a cyclic group, and find a generator. 1 1 1 1 1, −1, i, −i, √ (1 + i), √ (1 − i), √ (−1 + i), √ (−1 − i). 2 2 2 2 15.2. In Example 14.5 we constructed a table of logarithms to base 2 in the cyclic group F× 11 . Explain how the table can be used to show that 5 × 10 = 6 in this group. 15.3. Show that, for any positive integer n, the group (Zn , +) (the integers mod n under addition) is cyclic, and find a generator. Explain why the DLP is trivial in this case, however large n may be.

15.2 The general ElGamal cryptosystem The ElGamal systems described in Chapter 14 can be extended to any given cyclic group (G, ∗). Users of the general system are assumed to know that G is cyclic, and that a certain specified element g ∈ G is a generator. It is also assumed that they express plaintext and ciphertext messages as elements of G in some standard way.

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In the general ElGamal cryptosystem, each user, such as Bob, chooses a ′ private key b′ ∈ N, and computes his public key b ∈ G by the rule b = g b . When Alice wishes to send Bob a message m ∈ G she chooses a token t ∈ N, and applies the encryption function Eb (m, t) = (g t , m ∗ bt ). Bob receives ciphertext in two parts: the first part is the ‘leader’ ℓ = g t , and the other part is the encrypted message c = m ∗ bt . Bob’s decryption function is ′ Db′ (ℓ, c) = c ∗ (ℓ−1 )b . Using essentially the same algebra as in Lemma 14.7, it is easy to check that Db′ (Eb (m, t)) = m for all m ∈ G, and all t ∈ N: ′

Db′ (Eb (m, t)) = Db′ (g t , m ∗ bt ) = (m ∗ bt ) ∗ ((g t )−1 )b ′ = m ∗ bt ∗ ((g b )−1 )t = m ∗ bt ∗ (b−1 )t = m.

EXERCISES 15.4. Let (G, ∗) be a cyclic group of order 43, with generator g, and suppose Bob’s private key is 10. What is Bob’s public key, and what is his decryption function? If Alice wishes to send the message m ∈ G to Bob, and chooses t = 7, what ciphertext does Bob receive? Check that his decryption function correctly recovers m. (All working should be expressed in terms of the ‘variables’ g and m.) 15.5. Alice and Bob are experimenting with an ElGamal system based on the multiplicative group G = F× 17 , with generator g = 3. Bob’s public key is 13. Alice wishes to send the message m ∈ F× 17 to Bob. What encryption function should she use? Find Bob’s private key, write down his decryption function, and verify that it correctly recovers the message m. 15.6. In Exercise 15.3 we noted that (Zn , +) is cyclic group with generator 1. Show that in the corresponding ElGamal system b′ = b, and verify that the decryption function Db′ is the left-inverse of Eb .

15.3 Elliptic curves

241

15.3 Elliptic curves In the rest of the book we shall consider fields F with the property that 2x = 0 only if x = 0 (so that F = F2 , for example). This condition is expressed by the statement that F does not have characteristic 2. The reason for excluding fields with characteristic 2 is that the algebra takes a slightly different form in that case.

Definition 15.3 (Elliptic curve) Let F be a field which does not have characteristic 2. An elliptic curve over F is a set of ‘points’ (x, y) ∈ F 2 that satisfy an equation of the form y 2 = x3 + αx + β

(α, β ∈ F ),

together with one additional ‘point’, which is denoted by I and called the point at infinity. For cryptographic purposes we shall require that the field F is finite, but the same constructions can be used over any field F which does not have characteristic 2, and it is helpful to begin by looking at an example with F = R, the field of real numbers. In this case the ‘points’ that form the curve belong to the Euclidean plane R2 , and we can sketch the curve in the usual way. The resulting geometrical picture is very useful.

Example 15.4 Sketch the curve y 2 = x3 − x over R.

Solution The standard method of curve-sketching is to find points on the curve by choosing x ∈ R and calculating the value(s) of y such that y 2 = x3 −x. When x < −1 and when 0 < x < 1 the expression x3 − x is negative, and there are no corresponding values of y, since y 2 ≥ 0. When x = −1, 0, 1, x3 − x = 0 and y = 0 is the only possibility. Hence the points (−1, 0), (0, 0), and (1, 0) belong to the curve. Finally, for each remaining value of x (that is −1 < x < 0 and x > 1) the expression x3 − x is positive and there are two corresponding values of y. This means that the curve is symmetrical with respect to the x-axis: if (x, y) is on the curve, then (x, −y) is also on the curve. A sketch is shown in Figure 15.1. It must be remembered that I, the ‘point at infinity’, must also be considered. Geometrically, it is helpful to think of I as a point where all vertical lines meet; that is, an infinitely remote point in the vertical direction.

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Figure 15.1 A sketch of the curve y 2 = x3 − x in R2 Similar calculations can be used when the field is finite, but of course there is no sensible way of ‘sketching’ the curve.

Example 15.5 Find all the points on the curve y 2 = x3 + x over F17 . Solution As in the previous example, we consider each value of x and solve the resulting equation for y. For example, when x = 1 we require y 2 = 2, and in F17 this has two solutions, y = ±6. (Note that −6 = 11 here.) For each x ∈ F17 there are 0, 1, or 2 possible values of y: x= y= x= y=

0 1 2 0 ±6 −

3 4 ±9 0

5 6 7 − ±1 −

10 11 12 13 14 15 16 − ±4 − 0 ±2 − ±7

8 9 − − .

The calculation gives 15 points which, together with I, the point at infinity, comprise the curve. We shall now explain how an operation ∗ can be defined so that the points of an elliptic curve over a field F form an abelian group. The definition is based on a geometrical construction and is easy to visualize in the case when F = R. But the rules of algebra are the same in any field, so we can translate the construction into familiar coordinate geometry and apply it quite generally.

15.3 Elliptic curves

243

We begin by specifying that the point at infinity I is the identity element, so P ∗ I = I ∗ P = P for all points P . The inverse of a point P = (x, y) is P −1 = (x, −y): geometrically speaking, P −1 is the reflection of P in the x-axis. Note that if P is on the curve, so is P −1 . Also, if Q is a point of the form (x, 0) then Q−1 = Q, so Q ∗ Q = I. The crucial part of the construction is the definition of P1 ∗ P2 , which depends on the fact that the right-hand side of the equation is a polynomial of degree 3. It follows that a straight line y = λx + μ will generally meet the curve in three points. We define P1 ∗ P2 = S if and only if the points P1 , P2 and S −1 are collinear. Figure 15.2 shows three collinear points P1 , P2 , S −1 and the point S = P1 ∗ P2 .

S −1

P1

P2

S

Figure 15.2 Illustrating the group operation We now translate this construction into coordinate geometry. That is, we find equations for the coordinates of S = P1 ∗ P2 , given the coordinates of P1 and P2 .

Theorem 15.6 Suppose the points P1 = (x1 , y1 ) and P2 = (x2 , y2 ) belong to an elliptic curve y 2 = x3 + αx + β over a field F which does not have characteristic 2. Then the point S = P1 ∗ P2 is determined by the following rules: (i) if x1 = x2 and y1 = −y2 then S = I;

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15. Elliptic curve cryptography

(ii) in all other cases the coordinates (xS , yS ) of the point S are given by xS = λ2 − x1 − x2

yS = λ(x1 − xS ) − y1 ,

where λ is defined by λ = (y2 − y1 )(x2 − x1 )−1 λ = (3x21 + α)(2y1 )−1

if x1 = x2 ;

if x1 = x2 , y1 = y2 .

Proof (i) This is the situation when P2 = P1−1 , as defined above. (ii) Suppose that x1 = x2 and the line through P1 and P2 is given by the equation y = λx + μ. Then y1 = λx1 + μ

and

y2 = λx2 + μ,

so that λ = (y2 − y1 )(x2 − x1 )−1 ,

μ = λx1 − y1 .

This line meets the curve y 2 = x3 + αx + β at the points where x satisfies (λx + μ)2 = x3 + αx + β,

or x3 − λ2 x2 + (α − 2λμ)x + (β − μ2 ) = 0.

We know that two of the roots of this equation are x1 and x2 , and since the sum of the roots is λ2 , there is a third root xS given by xS = λ2 − x1 − x2 . Let yS = −(λxS +μ) so that the point S −1 = (xS , −yS ) is on the line y = λx+μ, and S = (xS , yS ) is the required point P1 ∗ P2 . Eliminating μ gives yS = λ(x1 − xS ) − y1 . If x1 = x2 the definition of λ will not work, because x2 − x1 = 0 has no inverse. In fact if x1 = x2 then y12 = y22 , so there are two possibilities, y1 = y2 or y1 = −y2 . The second possibility has already been dealt with (case (i)).

If x1 = x2 and y1 = y2 then P1 = P2 . In this case, we must determine the line y = λx + μ that meets the curve in two coincident points. In coordinate geometry over R we call this a tangent to the curve, and determine its slope λ by calculus. Because the curve has an algebraic equation, the same results hold in any field F , and the relevant value of λ is as given in the statement of the theorem. The rest of the algebra is as before, with this new value of λ.

15.4 The group of an elliptic curve

245

Example 15.7 In Example 15.5 we found that the points P1 = (1, 6) and P2 = (11, 4) belong to the curve y 2 = x3 + x over F17 . Calculate the coordinates of S = P1 ∗ P2 and T = P1 ∗ P1 .

Solution Taking P1 = (1, 6) and P2 = (11, 4), the coordinates of S = P1 ∗ P2 are given by λ = (4 − 6) × (11 − 1)−1 = (−2) × (10)−1 = −2 × 12 = −24 = 10, xS = 102 − 1 − 11 = 88 = 3,

yS = 10(1 − 3) − 6 = −26 = 8.

Thus S = (3, 8). To find T = P1 ∗ P1 we use the alternative form of λ: λ = (3 × 12 + 1) × (2 × 6)−1 = 4 × 10 = 40 = 6, xT = 62 − 1 − 1 = 34 = 0,

yT = 6(1 − 0) − 6 = 0.

Thus T = (0, 0).

EXERCISES 15.7. Consider the curve y 2 = x3 + x over F17 discussed in Examples 15.5 and 15.7. Show that (1, 6) generates a subgroup of order 4. 15.8. Find explicitly all the points on the elliptic curve y 2 = x3 + x over F13 . (There are 20 of them.) Calculate the coordinates of the points (3, 2) ∗ (5, 0),

(2, 6) ∗ (2, 6).

15.9. Show that a point P = (x, y) on an elliptic curve has order 2 (that is, P ∗ P = I) if and only if y = 0. 15.10. Taking F = R, derive the formula for λ given in Theorem 15.6 for the case P1 = P2 .

15.4 The group of an elliptic curve The ∗ operation endows an elliptic curve with the structure of an abelian group. The relevant properties are almost self-evident, with the exception of the associative law: (A ∗ B) ∗ C = A ∗ (B ∗ C). Since we have obtained explicit formulae for the group operation, the associative law can, if necessary, be checked by

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some rather tedious algebra. (The reader who wants to ‘really’ understand why it is true is advised to study a more theoretical account of elliptic curves.) Our aim here is to explain how the group of an elliptic curve can be used in practice, specifically as the basis for an ElGamal cryptosystem. In order to do this, we need to identify a suitable cyclic group, which may be a proper subgroup of the full group, and a generator for it. A very simple example follows.

Example 15.8 For the curve y 2 = x3 + 2x + 4 over F5 , find a cyclic subgroup and a generator of it. Solution We can tabulate the points on the curve in the usual way: x= y=

0 1 2 3 ±2 − ±1 −

4 ±1

Thus, remembering I, there are seven points. Since 7 is a prime number, the full group must be cyclic, and any element except I is a generator. More generally, it would be very useful if we could determine the size and structure of the group of any elliptic curve over a finite field. Sadly, this requires a substantial amount of theory and some nontrivial calculations. But mathematicians have succeed in finding many examples that are suitable for use in practice, and the basic principles are easy to understand.

Lemma 15.9 Let GE be the group of points on an elliptic curve E : y 2 = x3 + αx + β over a finite field F . Define m1 = the number of roots of x3 + αx + β = 0 in F ; m2 = the number of x ∈ F for which x3 + αx + β is a non-zero square in F .

Then

|GE | = m1 + 2m2 + 1.

Proof For each x ∈ F we count how many points (x, y) belong to E. Since F is a field, the equation y 2 = x3 + αx + β has at most two solutions. If y = θ is a solution then y = −θ is also a solution, so the number of distinct solutions is 0 or 2 unless θ = 0, when there is just one solution.

15.4 The group of an elliptic curve

247

In other words there are two solutions when x3 +αx+β is a non-zero square in F , one solution when x3 +αx+β = 0 in F , and no solutions when x3 +αx+β is not a square in F . Adding 1 for the point at infinity, we have the result. The lemma shows that when F = Fp , the largest possible value of |GE | is 2p + 1, which would occur if m2 = p. In fact, one would expect that only about half the values of x ∈ Fp are such that x3 + αx + β is a square, so that |GE | will be approximately equal to p. Using this idea, Hasse proved in 1933 (long before elliptic curves became part of cryptography) that √ √ p + 1 − 2 p ≤ |GE | ≤ p + 1 + 2 p. In the most favourable situation, |GE | itself is a prime. Then the entire group is a cyclic group, and can be used as framework for systems based on the ElGamal formulae. For example, this happens when E is the curve y 2 = x3 + 10x + β

over Fp ,

where p

= 2160 + 7 = 14615016373309029118203684832716283019655932542983

β

= 1343632762150092499701637438970764818528075565078.

It has been shown that |GE | = 14615016373309029118203683518218126812711137002561 = p − 13144981562069447795540422, and it is easy to check (with MAPLE) that |GE | is a prime number. It follows that GE is a cyclic group, and any non-identity element is a generator. So if we follow the prescription described above, we have the basis for cryptosystem. (Note that |GE | differs from p by a number with 26 digits, whereas p has 53 digits, in accordance with Hasse’s theorem.) Although this favourable situation cannot be expected to occur very often, in practice it is just as useful to be able to find a large prime dividing |GE |. In that case we have a large cyclic subgroup of GE .

EXERCISES 15.11. Verify explicitly that (2, 1) is a generator for the group obtained in Example 15.8.

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15.12. Consider elliptic curves of the form y 2 = x3 + x + β over F11 . Find three values of β for which m1 (Lemma 15.9) is 0, 1, 3, respectively. 15.13. Taking β = 6 in the previous exercise, show that the group of the curve is cyclic, and find a generator for it. 15.14. Let p be an odd prime. Show that the group of the elliptic curve y 2 = x3 + x over the field Fp has even order. Find the number m1 for this group, distinguishing the cases p = 4s + 1 and p = 4s + 3. 15.15. Any group of order 20 has a cyclic subgroup of order 5. [You are not asked to prove this, but if you are familiar with elementary group theory you may wish to do so.] Determine this subgroup explicitly for the curve described in Exercise 15.8.

15.5 Improving the efficiency of exponentiation From the cryptographic perspective, it remains to consider the problems that arise when we try to implement a cryptosystem based on an elliptic curve. The most costly operations in the ElGamal scheme are the exponentiations ′ – calculating the powers such as g t and (ℓ−1 )b that occur in the encryption and decryption functions. In order to ensure confidentiality the exponents must be numbers with many digits, and the exponentiations, although feasible by the repeated squaring algorithm, are by no means trivial. In fact, that is the main reason why the ElGamal system is commonly used only to distribute keys for a symmetric key system such as AES, in which the calculations are less costly. (Similar remarks apply to RSA, where exponentiation is also a major part of the system.) The problem of finding good algorithms for exponentiation is therefore significant. It is easy to see that the repeated squaring algorithm is not optimal: for example, it finds g 15 by calculating g 2 = g ∗ g, g 12 = g 4 ∗ g 8 ,

g4 = g2 ∗ g2, g 14 = g 2 ∗ g 12 ,

g8 = g4 ∗ g4, g 15 = g ∗ g 14 .

This procedure involves the sequence of powers 1, 2, 4, 8, 12, 14, 15, which has the property that each term in the sequence except the first is the sum of two (possibly the same) terms that come before it. Any sequence with this property that ends in 15 will produce the the required result.

15.5 Improving the efficiency of exponentiation

249

Definition 15.10 (Addition chain) An addition chain of length r for the positive integer n is a sequence of positive integers x0 = 1, x1 , x2 , . . . , xr = n such that, for i = 1, 2, . . . , r there exist xj and xk such that xi = xj + xk

(0 ≤ j ≤ k < i).

Clearly, shorter addition chains for n lead to better methods for calculating g n . For example, the repeated squaring method for g 15 corresponds to the addition chain of length 6 given above, but there is a shorter addition chain, with length 5: 1, 2, 3, 6, 12, 15. This corresponds to the multiplications g 2 = g ∗ g,

g3 = g ∗ g2,

g 12 = g 6 ∗ g 6 ,

g6 = g3 ∗ g3,

g 15 = g 3 ∗ g 12 .

A further improvement can be made when inversion is a trivial operation. As we shall explain in the next section, this holds true when G is the group of an elliptic curve. In such cases division (multiplication by a power of g −1 ) is no more costly than multiplication by g, which motivates the following definition.

Definition 15.11 (Addition-subtraction chain) An addition-subtraction chain of length r for the positive integer n is a sequence of positive integers x0 = 1, x1 , x2 , . . . , xr = n such that, for i = 1, 2, . . . , r there are xj and xk such that xi = ±xj ± xk

(0 ≤ j ≤ k < i).

In other words, each term in the sequence except the first is the sum or difference of two terms that come before it. The technique of exponentiation using an addition-subtraction chain is best illustrated by an example.

Example 15.12 Find the optimum method of calculating g 31 . Solution The repeated squaring algorithm uses the addition chain 1,2,4,8,16, 24,28,30,31, which has length 8. It is fairly easy to spot an addition chain of length 7: 1, 2, 3, 5, 10, 11, 21, 31, and some rather tedious analysis will confirm that it is the shortest possible.

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However, if subtractions are allowed there is an obvious chain of length 6: 1, 2, 4, 8, 16, 32, 31, and this is optimal. The corresponding method of calculating g 31 is g 2 = g ∗ g, g 4 = g 2 ∗ g 2 , g 8 = g 4 ∗ g 4 , g 16 = g 8 ∗ g 8 ,

g 32 = g 16 ∗ g 16 ,

g 31 = g −1 ∗ g 32 .

A useful technique for finding a good addition-subtraction chain is based on the non-adjacent form of an integer (Exercise 15.18). It leads to an algorithm for exponentiation that is about 10% better than repeated squaring, on average.

EXERCISES 15.16. Write down the addition chain for 127 used in the repeated squaring algorithm. This is a chain with length 12. Show that it is not optimal by finding an addition chain with length 10 for 127. 15.17. Show that the computation of g 127 can be shortened further if addition-subtraction chains are allowed. 15.18. Consider representations of an integer in the form  ci 2 i ci ∈ {−1, 0, 1}.

Such a representation is said to be a non-adjacent form or NAF if ci ci+1 = 0 for all i ≥ 0. Find a NAF for 55 and explain how it can be used to calculate g 55 .

15.19. Show that every integer has a NAF. [Hint: start from the standard binary representation.] Show also that the NAF is unique. 15.20. Why are addition-subtraction chains not useful for the calculation of g n when g is an element of F× p?

15.6 A final word Elliptic curve cryptography is a rapidly growing area of research, and it is possible that future developments will change the picture quite dramatically. Here is a summary of the current state of the art.

15.6 A final word

251

• The group of an elliptic curve can be used as the basis for a cryptosystem of the ElGamal type. By making suitable adjustments, elliptic curves can also be used in many other areas of cryptography. • The ElGamal functions can be calculated fairly efficiently, but the cost of exponentiation (in particular) imposes some constraints in practice. • It is possible to break an elliptic curve system if there is a method of solving the DLP on the group of the curve but, in general, no such method is known. Other forms of attack may be possible. We conclude with an explicit example of how the ElGamal formulae can be applied to the group of an elliptic curve. First, we must decide how to represent the elements of the group. For a curve E defined over a prime field Fp , an element of GE is a pair (x, y) with x, y ∈ Fp . So we can regard x and y as integers in the range 0 ≤ x, y ≤ p − 1. Furthermore, when the righthand side of the equation is a non-zero square there are exactly two values of y that satisfy the equation, and they can be written uniquely as ±θ, where θ satisfies 1 ≤ θ ≤ 12 (p − 1). Since θ is determined by E, in order to store (x, y) it is only necessary to store x as an element of Fp , together with a single bit that determines whether the relevant sign is + or −. Incidentally, this observation justifies the use of addition-subtraction chains for exponentiation, since inversion in GE is a trivially easy operation, the inverse of (x, y) being (x, −y). Let E denote the curve y 2 = x3 + x + 4

(x, y ∈ F23 ).

We shall use the notation x+ and x− for the points (x, y) with y = ±θ, 1 ≤ θ ≤ 11: for example, 0+ stands for (0, 2) and 0− stands for (0, −2). Substituting x = 0, 1, 2, . . . , 22 in turn, we find that x3 + x + 4 is never zero, so that m1 = 0, and it is a square when x = 0, 1, 4, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 22, so that m2 = 14. Hence the order of the group GE is 2 × 14 + 1 = 29. Since this number is prime, the group is cyclic, and any element except I can be taken as the generator g. Conveniently, the group GE has the right number of symbols to represent the english alphabet, extended to include the comma and full stop. Although this number is far too small to provide security in serious applications, it can be used to send messages that are unintelligible to the vast majority of people. To do this, we need to establish a ‘standard’ correspondence between the 29 elements of the group and the symbols of the extended english alphabet. Here is a suitable correspondence.

252

15. Elliptic curve cryptography

I ⊔

0+ A

0− B

1+ C

1− D

4+ E

4− F

7+ G

7− H

8+ I

8− J

9+ K

9− L

10+ 10− 11+ 11− 13+ 13− 14+ M N O P Q R S

14− 15+ 15− 17+ 17− 18+ 18− 22+ 22− T U V W X Y Z , . Suppose I am using GE with generator g = 0+, and my public key is b = 7−. If you have read this book carefully, you will be able to construct a table of powers of g. i 2 3 4 5 6 7 8 9 10 11 12 13 14 i g 13− 11+ 1− 7− 9+ 15+ 14+ 4+ 22+ 10+ 17+ 8− 18+ Since g 29−i is the inverse of g i , and the group is cyclic, this table is sufficient for all calculations in GE . In particular, you will quickly see that my private key is b′ = 5. Then, if you intercept some ciphertext intended for me, say (9+, 15−) (11+, 4−) (0+, 18+) (7−, 1+) (14+, 4−) (15+, 7+) (13−, 18+) (1−, 22+) you can apply my decryption function (ℓ, c) → c ∗ ℓ−5 and obtain 14 −

7−

4+

I

4+

10 −

1−

22−

which is definitely THE⊔END.

EXERCISES 15.21. The message above was encrypted using a different value of the token t for each symbol. Find these values. 15.22. Karen has agreed to use ElGamal cryptosystem based on the group GE defined above, with the ‘standard’ representation of extended english. She has chosen the generator h = 4−, and her public key is k = 9+. I have sent her the message (7−, 7+) (8−, 9+) (18−, 4−) (10+, 0+) (1−, 8−) (15−, 7+) (8−, 18−) (17−, 10+) (7+, 4−) (8−, 4−) . What does it say?

15.6 A final word

253

Further reading for Chapter 15 There are several books on the mathematical theory of elliptic curves, at various levels of sophistication. From the cryptographic perspective there are two fundamental results: Hasse’s theorem on the order of GE (Section 15.4), and a theorem that says GE can be expressed as the product of at most two cyclic groups. These results are discussed in the books by Silverman [15.5] and Washington [15.6], among others. The rapidly developing field of elliptic curve cryptography is surveyed in two books by Blake, Seroussi, and Smart [15.1, 15.2]. These books cover many of the implementation issues including the cost of exponentiation (Section 15.5). Further details on addition chains, the NAF, and so on, can be found in the survey by Gordon [15.3], and the famous tome by Knuth [15.4]. 15.1 I. Blake, G. Seroussi, N. Smart. Elliptic Curves in Cryptography. Cambridge University Press (1999). 15.2 I. Blake, G. Seroussi, N. Smart. Advances in Elliptic Curve Cryptography. Cambridge University Press (2005). 15.3 D.M. Gordon. A survey of fast exponentiation algorithms. J. Algorithms 27 (1998) 129-146. 15.4 D.E. Knuth The Art of Computer Programming II - Semi-numerical Algorithms. Addison-Wesley (third edition, 1997). 15.5 J.H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag (1986). 15.6 L.C. Washington Elliptic Curves: Number Theory and Cryptography. Chapman and Hall / CRC Press (2003).

Answers to odd-numbered exercises

Answers to the odd-numbered exercises are given below. In some cases the ‘answer’ is just a hint, in others there is a full discussion. A complete set of answers to all the exercises, password-protected, is available to instructors via the Springer website. To apply for a password, visit the book webpage at www.springer.com or email [email protected]. Chapter 1 ‘Canine’ has six letters and ends in ‘nine’. The second message has two possible interpretations. 1.3. The mathematical bold symbols A and B. 1.5. This exercise illustrates the point that decoding a message requires the making and testing of hypotheses. Here the rules are fairly simple, but that is not always so. In the first example, it is a fair guess that the numbers represent letters, and the simplest way of doing that is to let 1, 2, 3, . . . represent A, B, C, . . . . The number 27 probably represents a space. Testing this hypothesis, we find the message GOOD⊔LUCK. The second example has the same number of symbols as the first, and each is represented by a word with 5 bits. How is this word related to the corresponding number in the first example? 1.7. s1 s2 and s3 s1 are both coded as 10010. 1.9. In both cases S is the 27-symbol alphabet A. In the first example T = {1, 2, . . . , 27}, and the coding function uses only strings of length 1. In the second example T = B, and the coding function S → B∗ is an injection into the subset B5 of B∗ . 1.11. SOS; MAYDAY. 1.1.

256

Codes

1.13. Thenumber of ways of choosing 2 positions out of 8 is the binomial num ber 82 = (8 × 7)/2 = 28. Hence at most 28 symbols can be represented in the semaphore code. 1.15. Using words of length 2 there are only 4 possible codewords, so we need words of length 3, where we can choose any 6 of the 8 possibilities, say 1 → 001 2 → 010 3 → 011 4 → 100 5 → 101 6 → 110. With this code, if one bit in a codeword is wrong, then the result is likely to be another codeword: for example, if the first bit in 110 is wrong, we get the codeword 010. This problem cannot be avoided if we are restricted to using words of length 3. In order to overcome the problem we must use codewords with the property that any two differ in at least two bits. In that case, if one bit in any codeword is in error, then the result is not a codeword, and the error will be detected. This can be arranged if we use codewords of length 4, for example 1 → 0000 2 → 1100 3 → 1010 4 → 1001 5 → 0110 6 → 0101. 1.17. No, because the message refers to ENGLAND, which did not exist in Caesar’s time. Also it is written in English. Chapter 2 s3 s4 s2 s1 s4 s2 s3 s1 . The new code is s1 → 10, s2 → 1, s3 → 100. Clearly it is not prefixfree, since 1 is a prefix of both 10 and 100. However, it can be decoded uniquely by noting that each codeword has the form 1 followed by a string of 0’s Alternatively decoding can be done by reading the codeword backwards. If the last bit is 1, the last symbol must be s2 . If it is 0, looking at the last-but-one bit enables us to decide if the last symbol is s1 or s3 . Repeating the process the entire word can be decoded uniquely. For example, 110101100 decodes as s2 s1 s1 s2 s3 . 2.5. The code can be extended by adding words such as 011, 101, without losing the PF property. 2.7. 128. 2.9. (i) 00, 101, 011, 100, 101, 1100, 1101, 1110; (ii) 0, 100, 101, 1100, 1101, 1110, 11110, 11111. 2.11. In part (i) take T = {0, 1, 2}; then the codewords could be 00 and any 12 of the 18 words 1 ∗ ∗, 2 ∗ ∗. 2.1. 2.3.

Answers to odd-numbered exercises

257

2.13. The parameters n1 , n2 , n3 , n4 must satisfy n1 + n2 + n3 + n4 = 12,

n2 n3 n4 n1 + + + ≤ 1. 2 4 8 16

These equations imply that 7n1 + 3n2 + n3 ≤ 4, so n1 = 0 and n2 ≤ 1. Now it is easy to make a list of the possibilities: n2 : 1 1 0 n3 : 1 0 4 n4 : 10 11 8

0 0 0 0 3 2 1 0 9 10 11 12

.

2.15. The coefficient of x4 in Q2 (x) is the number of S-words of length 2 that are represented by T -words of length 4. These 25 words are all the words si sj with i, j ∈ {2, 3, 4, 5, 6}. 2.17. Use the fact that Qr (x) = Q1 (x)r . Chapter 3 3.1. 3.3.

3.5.

Occurrences of: a, about 60; ab, about 18. This is a deterministic source. The probability distribution associated with each ξk is trivial. For example, Pr(ξ4 = 3) = 1, Pr(ξ4 = n) = 0 for n = 3. Suppose the word-lengths are x1 ≤ x2 ≤ x3 , so the average is L = x1 α + x2 β + x3 (1 − α − β). The KM condition is 1 1 1 + x2 + x3 ≤ 1. 2 x1 2 2

3.7. 3.9.

The least possible value of x1 is 1. If also x2 = 1 there is no possible value for x3 . If x2 = 2, we must have x3 = 2. So we get the ‘obvious’ solution x∗1 = 1, x∗2 = 2, x∗3 = 2, which gives L = 2 − α. Any other solution must have xi ≥ x∗i , with at least one strict inequality, and since L is an increasing function of x1 , x2 , x3 , the average word-length will also be greater. 0. Start by proving that U H(u1 /U, u2 /U, . . . , um /U ) = U log U +

m 

ui log(1/ui ).

i=1

3.11. h′ (x) = (1/ ln 2) log((1 − x)/x). This is zero when (1 − x)/x = 1, that is, x = 21 . As x tends to 0 or 1, log((1 − x)/x) tends to ±∞.

258

Codes

3.13. The SF rule says that the word-lengths are such that x1 is the least integer such that 2x1 ≥ (1/0.25) = 4, that is, x1 = 2, and so on. The results are x1 = 2, x2 = 4, x3 = 3, x4 = 5, x5 = 3, x6 = 2. The average word-length is 2.7, and the entropy is H(p) ≈ 2.42. 3.15. The probabilities at each stage are as follows (no attempt has been made to make them increase from left to right). The new entry in each line is in bold. 0.25 0.25 0.25 0.25

0.1

0.15 0.05 0.15

0.15 0.3 0.3

0.2 0.2 0.2

0.25 0.25 0.25 0.45 0.45

0.55 1.0

Using rule H2 gives the following codewords (minor variations are possible): 01, 0011, 000, 0010, 10, 11, with L = 2.45. 3.17. The entropy is 2.72 approximately. The SF rule gives codewords of lengths 3, 3, 3, 4, 4, 4, 4 with average word length LSF = 3.4. This satisfies the condition H(p) ≤ LSF < H(p) + 1. The sequence of probability distributions generated by the Huffman rule is as follows (the probabilities have not been re-ordered on each line). 0.2 0.2 0.2 0.2 0.2 0.6 1.0

0.2 0.2 0.2 0.2 0.4

0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.4 0.4 0.4

0.1

Using H2 the codewords are 00, 010, 011, 100, 101, 110, 111. (Several choices are possible, but all of them will produce a code with one word of length 2 and six of length 3.) The average word-length is Lopt = 2.8. 3.19. Use Lemma 3.17. 3.21. Using tree diagrams a few trials produces the code 0, 10, 11, 12, 20, 21, 22, which has average word-length 1.8. Since the probabilities are all multiples of 1/10, the average word length of any such code is a number of the form m/10. However the entropy with respect to encoding by a ternary alphabet is H3 (p) = H(p)/ log2 3 ≈ 1.72. Hence the word-length 1.8 must be optimal.

Answers to odd-numbered exercises

259

Chapter 4 4.1. 4.3.

k ≥ rℓ + ℓ − 1. The entropy of the given distribution is approximately 0.72. For the obvious code A → 0, B → 1, clearly L1 = 1. For blocks of length 2 the probabilities are 0.64, 0.16, 0.16, 0.04, and a Huffman code is AA → 0, AB → 10, BA → 110, BB → 111. This has average word-length 1.56 and L2 /2 = 0.78. For blocks of length 3 the probabilities and a Huffman code are AAA AAB 0.512 0.128 0 100

ABA BAA 0.128 0.128 101 110

ABB BAB BBA BBB 0.032 0.032 0.032 0.008 11100 11101 11110 11111.

Thus L3 = 2.184 and L3 /3 = 0.728. This suggests that the limit of Ln /n as n → ∞ is the entropy, 0.72 approximately. 4.5. H(p) ≈ 2.446, H(p′ ) ≈ 0.971, H(p′′ ) ≈ 1.571. 4.7. p1 = [0.4, 0.3, 0.2, 0.1]. Not memoryless. 4.9. H ≤ H(p2 )/2 ≈ 1.26. 4.11. Use the hint given. 4.13. It is sufficient to consider the range 0 < x < 0.5, when the original distribution is [x2 , x(1 − x), x(1 − x), (1 − x)2 ] and the numerical values increase the order given. The first step is to amalgamate x2 and x(1 − x), giving the distribution [x, x(1 − x), (1 − x)2 ]. Since x > x − x2 the middle term is always one of the two smallest. The other one is x if 0 < x ≤ q and (1 − x)2 if √ q < x ≤ 0.5, where q is the point where x = (1 − x)2 . In fact, q = (3 − 5)/2 ≈ 0.382. In the first case the word-lengths of the optimal code are 3, 3, 2, 1, and in the second case they are 2, 2, 2, 2. Hence L2 (x) = 1 + 3x − x2 (0 < x ≤ q),

L2 (x) = 2 (q < x ≤ 0.5).

4.15. In binary notation 1/3 is represented as .010101 . . ., where the sequence 01 repeats for ever. In general a rational number is represented by an expansion that either terminates or repeats, depending on the base that is used. For example, the representation of 1/3 repeats in base 2 and base 10, but terminates in base 3. 4.17. It is enough to calculate the values of nP , since the average word-length  is L2 = 1 + P nP ≈ 4.44. Since H(p1 ) ≈ 1.685; H(p2 )/2 ≈ 1.495, the entropy does not exceed 1.495, whereas L2 /2 ≈ 2.22. 4.19. For X = x1 x2 . . . xr−1 xr let X ∗ = x1 x2 . . . xr−1 α, where α is the first element of S. The probabilities P (Y ) with Y < X can be divided into two sets: those with Y < X ∗ and those with X ∗ ≤ Y < X. 4.21. L2 ≈ 5.01. H = H(P ) ≈ 1.68 < L2 /2.

260

Codes

4.23. Check that the encoding rule creates dictionaries in Example 4.25. 4.25. The message begins BET⊔ON⊔TEN⊔ . . . . Chapter 5 5.1.



1−a b

a 1−b



.

Let p be the input to Γ1 , q the output from Γ1 and input to Γ2 , r the output from Γ2 . Then r = qΓ2 = pΓ1 Γ2 . Hence Γ is the matrix product Γ1 Γ2 . 5.5. Since en+1 = en + e − 2een it follows that en → 12 as n → ∞. When e = 0, we have en = 0 for all n, so the limit is zero. When e = 1 the value of en alternates between 0 and 1, so there is no limit. 5.7. q0 = 0.98p0 + 0.04p1 , q1 = 0.02p0 + 0.96p1. 5.9. t00 = 0.594, t01 = 0.006, t10 = 0.004, t11 = 0.396. H(t) ≈ 1.0517 and (as in Exercise 5.6) H(q) ≈ 0.9721, hence H(Γ ; p) ≈ 0.0796. 5.11. We have q = pΓ = [pc (1 − p)c 1 − c], from which it follows by direct calculation that H(q) = h(c) + ch(p). For the joint distribution t the values are pc, 0, p(1 − c) and 0, (1 − p)c, (1 − p)(1 − c). Thus (by the argument used the proof of Theorem 5.9) H(t) = h(p) + h(c), and 5.3.

H(Γ ; p) = H(t) − H(q) = h(p) + h(c) − h(c) − ch(p) = (1 − c)h(p). 5.13. (i) 1/27; (ii) 1; (iii) 3. 5.15. The channel matrix Γ is a 2N × 2 matrix with rows alternately 0 1 and 1 0. If the input source is [p1 , p2 , . . . , p2N ], the joint distribution t is given by t1,0 = 0, t1,1 = p1 , t2,0 = p2 , t2,1 = 0,

...

, t2N,0 = p2N , t2N,1 = 0.

Hence H(t) = H(p). It follows that H(p) − H(Γ ; p) = H(p) − H(p | q) = H(p) − H(t) + H(q) = H(q). So the maximum of H(p) − H(Γ ; p) is the maximum of H(q), which is 1. This means that the channel can transmit one bit of information about any input: specifically, it tells the Receiver whether the input is odd or even. 5.17. For each input x, H(q | x) = 2α log(1/α) + (1 − 2α) log(1/(1 − 2α)) = k(α) say. Hence H(q | p) = k(α), which is constant. The maximum of H(q) occurs when q = [1/4, 1/4, 1/4, 1/4], and so the capacity is 2 − k(α).

Answers to odd-numbered exercises

261

Chapter 6 6.1. 6.3. 6.5. 6.7. 6.9.

6.11. 6.13.

6.15. 6.17. 6.19.

One of the instructions S,E,W. Yes, because 100100 is more like 000000 than any other codeword. The first column is (1 − a)2 , (1 − a)b, (1 − a)b, b2 . c7 , c5 , c3 . The nearest codewords for z1 are 11000, 01100, 01010, 01001. So σ(z1 ) is certainly not 10001, and this event has probability zero. The nearest codewords for z2 are 11000, 10100, 10010, 10001. The rule is that one of them is chosen as σ(z2 ) with probability 1/4, so the probability that z2 is received and σ(z2 ) = 10001 is e/4. The nearest codewords for z3 and z4 do not include 10001, so (like z1 ) these contribute nothing. The nearest codewords for z5 are 01001, 10001, 11000. The rule is that one of them is chosen as σ(z5 ) with probability 1/3, so the probability that z5 is received and σ(z5 ) = 10001 is e/3. Hence the required probability is 7e/12. The MD rule would give σ(100) = 000, for example, which is clearly not the same as the maximum likelihood rule in this case. For each c ∈ C, there are 6 words that can result from one bit-error, so |N1 (c)| = 7. Any of the resulting 5 × 7 = 35 words can be corrected, so the number of words that cannot be corrected is 64 − 35 = 29. We require ρ = (log2 |C|)/6 to be at least 0.35. Thus log2 |C| ≥ 2.1, which means that |C| ≥ 5. For an example, see Exercise 6.12. n = 14. Suppose C is a maximal code for the given values of n and r. Then the neighbourhoods N2r (c) (c ∈ C) must completely cover Bn , otherwise there would be a word x ∈ Bn such that d(x, c) ≥ 2r + 1 for all c ∈ C, contradicting the maximality of C.

Chapter 7 7.1.

7.3.

For each c ∈ C there are two other codewords. The MD rule does not assign these codewords to c, nor the words at distance 1 from them. Hence F (c) contains at least 2 × (1 + 5) = 12 words. If p = (0.9, 0.1) we have q = (0.74, 0.26). For the ideal observer rule we need the conditional probabilities Pr(c | z) which can be calculated using the rule Pr(c | z)qz = Pr(z | c)pc , where Pr(z | c) = Γcz . The numbers are Pr(c = 0 | z = 0) ≈ 0.98, Pr(c = 0 | z = 1) ≈ 0.70, Pr(c = 1 | z = 0) ≈ 0.02, Pr(c = 1 | z = 1) ≈ 0.30. The ideal observer rule σ ∗ says that σ ∗ (z) = c when Pr(c | z) ≥ Pr(c′ | z) for all c′ ∈ C. In this case σ ∗ (0) = 0, σ ∗ (1) = 0: in other words, the Receiver always

262

7.5. 7.7.

Codes

decides that 0 was the intended codeword. Hence M0 = 0, M1 = 1, and the probability of a mistake is 0.9M0 + 0.1M1 = 0.1. 10e3 (1 − e)2 + 5e4 (1 − e) + e5 = 10e3 − 15e4 + 6e5 . Since |Rn | = 2, the information rate is 1/n, which tends to 0 as n → ∞. As in Exercise 6.10, the channel matrix has the form 00 · · · 0 11 · · · 1

1 fn

0 gf n−1

0 gf n−1

. . . 0 , . . . gn

where g = 1 − f . Thus the maximum likelihood rule gives σ(00 · · · 0) = 00 · · · 0, but σ(z) = 11 · · · 1 for all z = 11 · · · 1. A mistake occurs only when 11 · · · 1 is sent and all bits are transmitted wrongly. The probability of a mistake is therefore (1 − p)f n , which also tends to 0. 7.9. The code in Example 7.5 has parameters (6, 3, 3), and the new code has parameters (6, 3, 2). 7.11. Since e = 0.03, we have γ = 1 − h(e) ≈ 0.80. Thus the uncertainty is at least 0.1 n, where n is the length of the codewords. 7.13. Use the fact that H(q) ≤ H(q′ ) + H(q′′ ). 7.15. Let p = [p, 1 − p]. Since e = 0.5, q = [0.5, 0.5]. Hence the left-hand side of Fano’s inequality is h(p) − h(e) + h(q) = h(p). Since M = e = 0.5 (Example 7.2) the right-hand side is 1, and the inequality is just h(p) ≤ 1. In particular, when p = 0.5 there is equality. Chapter 8 8.1. 8.3.

C1 is linear but C2 is not, because (for example) 100 + 010 = 110 which is not in C2 . (a) A linear code with dimension k has 2k codewords, so if there are 53 students we require 2k ≥ 53. Thus k = 6 is the minimum possible value.

(b) Suppose that the codewords have length n, and the code allows for the correction of one error. Then the neighbourhoods N1 (c), consisting of a codeword c and the n words that can be obtained from c by changing one bit, must be disjoint. There are 26 neighbourhoods, so 26 (1+n) ≤ 2n , that is, n + 1 ≤ 2n−6 . The least value of n for which this is true is 10. 8.5. Since 0 + 0 = 0 and 1 + 1 = 0, the weight of x + y is equal to the number of places where x and y differ. This is equal to (number of places where x is 1 and y is 0) + (number of places where x is 0 and y is 1). The first term is equal to w(x) − w(x ∗ y) and the second term is equal to w(y) − w(x ∗ y). Hence w(x + y) = w(x) + w(y) − 2w(x ∗ y). In particular, if x and y both have even weight, so does x + y. This proves that the set of words of even weight is a linear code.

Answers to odd-numbered exercises

263

8.7.

Let I denote the k × k identity matrix. Then   I E= . 1 1 . . . 1 1

8.9. 8.11. 8.13. 8.15.

ρ = k/(k + 1), δ = 2. The matrix [1 1 1 . . . 1], with k + 1 columns. A suitable matrix is given in Exercise 8.17. It helps to note that columns 2,3,5,6 are the columns of the identity matrix (in scrambled order). This means that x2 , x3 , x5 , x6 are determined by x1 , x4 , x7 . Explicitly, we can write the equations in the form x2 x5 x3 x6

= x1 + x4 + x7 = x4 + x7 = x1 + x4 + x7 = x7 .

Thus there are 8 codewords, corresponding to the 23 choices for the bits x1 , x4 , x7 . For example, if x1 = 0, x4 = 0, x7 = 1 the equations say that x2 = 1, x3 = 1, x5 = 1 and x6 = 1. Since the columns of H are non-zero and all different, Theorem 8.10 implies that the minimum distance is 3, at least. Thus n = 7, k = 3, δ = 3. 8.17. Given z ′ = [111010]′, we find that Hz ′ = [110]′ . This is the second column of H, so there was an error in the second bit, and the intended codeword was 101010. 8.19. The given check matrix defines x3 , x4 , x5 in terms of x1 , x2 , by the equations x3 = x1 , x4 = x2 , x5 = x1 + x2 . The code words are obtained by giving the values 00, 01, 10, 11 to x1 , x2 and using the equations. So they are 00000, 01011, 10101, 11110. There are eight cosets and the syndromes and coset leaders are: 000 100 010 001 110 101 011 111 00000 00100 00010 00001 11000 10000 01000 01100. Suppose the received word is z = 11111. Then the syndrome Hz ′ is 001, which corresponds to the coset leader f = 00001. According to the rule, σ(z) = z + f = 11110, which is indeed a codeword. In the other cases the codewords are 11110, 10101, 11110. 8.21. The syndrome of z is 10000000′. It is not the zero word, or a column of H, so more than one bit-error has occurred. If two bit-errors have occurred, the syndrome must be the sum of two columns of H. The possiblities for the first four rows are (h1 , h9 ), (h2 , h10 ), and so on. The corresponding pairs in the bottom four rows are (h5 , h8 ), (h3 , h5 ), and so on. Checking these pairs we find the sixth pair is (h15 , h15 ) and h15 + h15 = 0000′ . This corresponds to (h6 , h14 ), so errors have occurred in bits 6 and 14.

264

Codes

Chapter 9 9.1.

9.3.

9.5.

The check matrix can be obtained by writing down the 15 columns corresponding to the binary representations of the numbers 1, 2, . . . , 15, in order. The number of codewords is 215−4 = 2048. Denote the given words by z1 , z2 , z3 . Then Hz1′ = [1010]′ , Hz2′ = [0111]′, Hz3′ = [1000]′ . Hence z1 has an error in the 10th bit, z2 has an error in the 7th bit, and z3 has an error in the 8th bit. The number of cosets is 2n /|C| = 2n /2n−m = 2m , where m is the number of rows of a check matrix. For a Hamming code n = 2m − 1. The syndromes of the n + 1 given words are all distinct. 1 + x + x2 represents the word 111. The other codewords are obtained by multiplying 1 +x+ x2 by f (x), where f (x) is any polynomial with degree less than 3, and reducing mod x3 −1. It turns out that there are only two possibilities 0 and 1 + x + x2 itself, so the code is {000, 111}. Since the codewords are defined by the equations x1 = x2 = x3 a suitable check matrix is   1 1 0 H= . 0 1 1

Yes. The code defined by the ideal 1 + x contains all words with weight 2 and hence all words with even weight. 9.9. Three of the 8 divisors, 1, x7 − 1, and 1 + x + x3 , are discussed in Example 9.14. The divisor 1 + x generates the code containing all words with even weight, and the divisor 1 + x2 + x3 generates a code equivalent to the Hamming code. The divisor (1 + x)(1 + x + x3 ) corresponds to h(x) = 1 + x2 + x3 , and the resulting check matrix defines a code with parameters (7, 3, 3). The divisor (1+x)(1+x2 +x3 ) is similar. The divisor (1 + x + x3 )(1 + x2 + x3 ) corresponds to h(x) = 1 + x, and the resulting check matrix defines a code with parameters (7, 1, 7), in other words the repetition code {0000000, 1111111}. 9.11. Since the Hamming code with word-length 15 = 24 − 1 has dimension 11 = 24 −1−4, the canonical generator g(x) must have degree 4 (Theorem 9.13). Taking g(x) = 1 + x + x4 , the complementary factor is 9.7.

h(x) = 1 + x + x2 + x3 + x5 + x7 + x8 + x11 . According to the rule given in the text, the corresponding matrix H has 4 rows and 15 columns, all of which are different, so it must define a code equivalent to the Hamming code. (Does the same result hold if we choose one of the other factors of degree 4?) 9.13. The usual method of evaluating a determinant by expanding in terms of the first row gives the result a33 − a31 + a25 − a30 + a26 − a23 , which factorizes as claimed.

Answers to odd-numbered exercises

265

9.15. The packing bound requires 16(1 + n + 21 n(n − 1)) ≤ 2n . The smallest integer for which this holds is n = 10. 9.17. The word 100101 01101 is a codeword but its cyclic shift 110010 10110 is not. 9.19. x3 − 1 = (1 + x)(1 + x + +x2 ). For dd = 3, the canonical generator is m1 (x) = 1 + x + x2 . Now refer to Exercise 9.5. 9.21. The Hamming code has parameters (15, 11, 3) so it corrects 1 error and has rate 11/15. The BCH code has parameters (15, 7, 5), so it corrects 2 errors, but has a lower rate 7/15. 9.23. Let α be as in Example 9.22, so that m1 (x) = 1 + x + x4 . Using the table of powers of α it can be verified that m3 (x) = 1 + x + x2 + x3 + x4 , m5 (x) = 1+x+x2 , m7 (x) = 1+x3 +x4 . Thus the canonical generators for the cases dd = 7 and dd = 9 have degrees 10 and 14 respectively, and the dimensions of the BCH codes are 5 and 1. The codes for dd = 11, 13, 15 are subcodes of the code for dd = 9, and in fact all these codes are simply the repetition code with just two words. Chapter 10 10.1. 10.3.

Use the addition formula, as in Example 4.7. It is convenient to write a b c instead of ∗ ! ? . The string becomes cbaacbbaaccbbaacbaac . An estimate of the probabilities for the firstorder approximation is obtained by counting the frequencies of the letters in the given string: pa = 8/20 = 0.4, pb = 6/20 = 0.3, pc = 6/20 = 0.3. Hence the entropy is 0.4 log3 (1/0.4) + 0.3 log3 (1/0.3) + 0.3 log3 (1/0.3) ≈ 0.991. Similarly, we can estimate the probabilities of the digrams. Those with non-zero frequency are paa = 4/19, pac = 4/19, pba = 4/19, pbb = 2/19, pcb = 4/19, pcc = 1/19.

10.5.

10.7.

From this we can calculate that the uncertainty (in bits per symbol) is approximately 0.776. The reduction in uncertainty is due to the predictability of the language. In particular the word !**? occurs four times in the given string. The symbols x and y both have probability 0.5 so U1 = 1. A simple calculation gives U2 ≈ 0.985. Since U is the infimum of the values Un it follows that U ≤ U2 < 1. LOTS⊔OF⊔PEOPLE⊔SUPPORT⊔MANCHESTER⊔UNITED.

266

Codes

The textish alphabet has more symbols than english, because numerals are often used, for example 4 replaces FOR. It is reasonable to suppose that textish has greater uncertainty than english, because there are fewer rules. 10.11. The key is 9. MATHEMATICS⊔IS⊔OFTEN⊔USEFUL . 10.13. In cycle notation, the encryption key is (⊔)(AROJNHCPKDUTSQMGI)(BE)(FL)(V)(W) (X)(Y)(Z) .

10.9.

The ciphertext is CRKKY EAOSCURY IORHHY. 10.15. An encryption key that is the same as the decryption key is a permutation σ such that σ = σ −1 . If there are no fixed letters (1-cycles), the cycle form of σ must consist entirely of 2-cycles. Let Tn denote the number of permutations of a set of size 2n that have n 2-cycles. Then Tn = (2n − 1)Tn−1 . Hence there are T13 = 25 × 23 × · · · × 3 × 1 = 7905853580625 possible keys. Although this a large number in everyday terms, if Eve has a reasonable computer she could carry out an exhaustive search quite quickly. Chapter 11 (ADOKG)(BECMI)(FRPLHTSQNJ). The key for decryption is therefore (AGKOD) (BIMCE)(FJNQSTHLPRF) . 11.3. THEINDEXOFCOINCIDENCEISAPOWERFULTOOL. 11.5. UW IL IC ER RL NL LG IZ NG OV TW CN IM KV FX VG FM MF ST . 11.7. Suppose the ciphertext digram is ab. Case 1: if a and b are in different rows and columns, then ab → xy, where a, b, x, y are the corners of a rectangle and a, x are in the same row. 11.9. [18 2]′[3 2]′ [12 22]′[17 17]′. 11.11. The block AMPLE corresponds to [1, 13, 16, 12, 5]′, which is encrypted as [17, 5, 2, 6, 13]′, so the required pair is (17, 5). Similar calculations for the other blocks result in different pairs. 11.13. The decryption function is x = γy + δ, where γ = α−1 and δ = −βα−1 . Since γ and δ can be calculated easily when α and β are known, we have a symmetric key system. If two plaintext-ciphertext pairs (x1 , y1 ) and (x2 , y2 ) are known, then the equations y1 = αx1 + β and y2 = αx2 + β can be solved for α and β. (There is an exception – what is it?) If the plaintexts are chosen to be x1 = 0 and x2 = 1, the solution is especially simple. 11.1.

Answers to odd-numbered exercises

267

Chapter 12 H(r) = 1, H(q) = 2, so H(r | q) is equal to H(p) − 1 ≈ 0.846439. It is fairly easy to find two meaningful words of the form xyx that are obtained from one another by a cyclic shift: for example DID and PUP. A ciphertext aba that encrypts one of these words also encrypts the other, with a different key. 12.5. The columns of Γ are not constant (see Theorem 12.6). 12.7. ELVIS⊔LIVES. 12.9. Suppose Eve knows m1 + m2 and conjectures that a segment y1 of m1 represents TUESDAY. If z is the corresponding segment of m1 + m2 the corresponding segment of m2 is y2 = y1 + z. If y2 is meaningful, it is likely that Eve’s conjecture is correct. 12.11. Use the decryption rule given in Theorem 12.10. 12.13. x′ = 1100, y ′ = 1111, so d(x′ , y ′ ) = 2. 12.15. The permutations are δ = (ADOKG)(BECMI) (FRPLHTSQNJ) and ρ = (AROJNHCPKDUTSQMGI)(BE)(FL). The double-locking procedure involves the application of the permutation ρ−1 δ −1 ρδ and it is easy to verify that it is not the identity permutation: for example, ρ−1 δ −1 ρδ (A) = D. 12.1. 12.3.

Chapter 13 13.1. 13.3.

The number of invertible elements is 12; 5−1 = 29. If xy = 1 (mod n), say xy = 1 + nk, then (n − x)(n − y) = n2 − n(x + y) + xy = 1 + n(n − (x + y) + k),

so n − y is the inverse of n − x. Hence the invertible elements occur in pairs, x and n − x (which must be distinct). So the number is even. 13.5. Here 3599 = 61 × 59, so φ(n) = 60 × 58 = 3480. Thus the private key is given by 31d = 1 (mod 3480), which is true when d = 3031. 13.7. By the Euclidean algorithm, gcd(15, 68) = 1 and 15−1 = 59. 13.9. The exponent has 10 decimal digits, so the crude estimate in the text gives an upper bound of 66. In fact the binary representation of 3578674567 has 31 binary digits, 18 of which 1’s, so the actual number is 31 + 17 = 48. 13.11. Using the fact that 1000 = 512 + 256 + 128 + 64 + 32 + 8, it turns out that 21000 = 27 (mod 47). 13.13. φ(15) = 8, so d = 3. It is easy to check that m9 = m (mod 15) for 1 ≤ m ≤ 14 (only the primes need be checked).

268

Codes

13.15. If n = pq and gcd(m, n) = 1 then (without loss of generality) we can take m to be a multiple of p, and trivially mK+1 = m (mod p). Also gcd(m, q) = 1 so mq−1 = 1 (mod q). Hence mK+1 = mkφ(n)+1 = mk(p−1)(q−1) × m = m (mod q). So if de = 1 (mod φ), mde = m + Aq. Since both m and mde are multiples of p, so is A. Hence mde = m (mod pq). 13.17. Given n = 4189 and φ = 4060, it follows that p + q = n − φ + 1 = 130. Hence p and q are the roots of x2 − 130x + 4189 = 0. Using the formula, we find p = 71, q = 59. Finally, d = e−1 (mod 4189) = 3023. 13.19. The fact that n is prime can be established relatively easily. If that is known, then φ(n) = n − 1 and the private key d is the inverse of e mod n − 1, which can be found easily using the Euclidean algorithm. Hence if c = me is an encrypted message addressed to the user, it can be decrypted easily by the formula m = cd . Chapter 14 14.1. 14.3. 14.5.

14.7.

14.9.

For each set of μy elements with hash value y there are μy (μy − 1)/2 collisions. If hk is not collision-resistant then given one pair (m1 , hk (m1 )) it is easy to find m′ with hk (m′ ) = hk (m), so hk (m′ ) is known. We have to show that the order of 3 is 16, in other words, the order is not 2, 4, or 8. We find 32 = 9, 34 = 13, 38 = 16. In this case, the other primitive roots are the odd powers of 3. ′ We require Da′ (Ea (m)) = m, that is (ma )a = m. This means aa′ = 1 mod p − 1, so gcd(a, p − 1) = 1. Similarly, gcd(b, p − 1) = 1. The doublelocking procedure works in this case because the functions commute. The following table shows n and 2n mod 19. 1 2

2 3 4 8

10 11 12 17 15 11

4 5 6 7 8 16 13 7 14 9

9 18

13 14 15 16 17 18 3 6 12 5 10 1

2 is primitive root, because it has order 18. Since 215 = 12 the logarithm of 12 is 15, and since 211 = 15 the logarithm of 15 is 11. 14.11. The result follows from the equation rα+β = rα rβ , remembering that α and β are integers in the range 1 ≤ α, β ≤ p − 1 and rp−1 = 1. 14.13. (ℓ, c) = (rt , xk t ) = (7777, 6532).

Answers to odd-numbered exercises

269

14.15. Since A’s private key is xA = 33, her public key is yA = 2xA = 233 (mod 149). If necessary, this can be calculated by hand: 22 = 4, 24 = 16, 28 = 256 = 107, 216 = 1072 = 125, 232 = 1252 = 129, 233 = 109. 14.17. Applying the test given in Theorem 14.13, we find a|i| ij = 62 × 24 = 1,

r|m| = 56 = 8,

so the signature is not valid. 14.19. The verification test is a|i| ij = r|m| , where a, r, m are known. If i is given a specific value then ij is known, and finding j is a DLP in F× p. On the other hand if j is given a specific value, i occurs twice in the resulting equation, and the problem is (apparently) more difficult. Chapter 15 15.1.

15.3. 15.5.

15.7. 15.9.

15.11. 15.13.

15.15. 15.17. 15.19.

15.21.

The first four numbers have orders 1, 2, 4, 4 respectively, and so they are not generators. All the other numbers are generators. (If you know that the numbers are the eighth roots of unity in C, this is trivial.) For any n, 1 is a generator, and 1 + 1 + · · · + 1 (i times) is i. Hence, given h, the solution to the DLP 1 + 1 + · · · + 1 (i times) = h is i = h. Alice chooses t and sends (3t , m × 13t ). Bob’s private key b′ is the ′ ′ solution of the DLP 3b = 13 in F× 17 , and by trial and error b = 4. So Bob decrypts using the rule (m × 13t ) × ((3t )−1 )4 , which reduces to m, for all t. (1, 6) ∗ (1, 6) = (0, 0) (Example 15.7), and (0, 0) ∗ (0, 0) = I, by case (i) of Theorem 15.6. So (1, 6) has order 4. For any x, case (i) of Theorem 15.6 says that (x, 0) ∗ (x, 0) = I. Conversely, if y = 0 then (x, y) ∗ (x, y) is determined by case (ii) and is not I. Let f = (2, 1). Then f 2 = (0, 3), f 3 = (4, 1), f 4 = (4, 4), f 5 = (0, 2), f 6 = (2, 4), f 7 = I. When β = 6 the relevant numbers are m1 = 0 (by the previous exercise) and m2 = 6 (by explicit calculation). Hence the group has order 13, which is prime, and so it is cyclic. Any element except I is a generator; for example (−1, 2). One element of order 5 is (4, 4) (there are others). The addition-subtraction chain 1, 2, 4, 8, 16, 32, 64, 128, 127 has length 8. If the binary representation has two adjacent 1’s, say ci−1 = ci = 1, the identity 2i = 2i+1 − 2 × 2i−1 gives a new representation with c′i−1 = −1, and c′i = 0. The result follows by applying this transformation recursively. The values of t are: 6 3 1 5 8 7 2 4.

Index

addition chain, 248 addition-subtraction chain, 249 Advanced Encryption Standard, 203 AES, 203 alphabet, 4 arithmetic code, 63 ASCII, 3 authentication, 221 average word-length, 30 b-ary, 5 BCH code, 156 binary, 5 binary asymmetric channel, 76 binary erasure channel, 80 binary symmetric channel, 74 bit, 2 bit-error probability, 74 BSC, 74 Caesar’s system, 170 canonical generator, 149 capacity, 82 channel, 74 channel matrix, 74 characteristic 2, 241 check bits, 130 check matrix, 129 chosen plaintext attack, 187 ciphertext, 172 code, 5 codeword, 5 collision, 223

collision-resistant, 223 concatentation, 6 conditional entropy, 79 coset, 136 coset leader, 137 cryptosystem, 179 cumulative probability function, 63 cyclic code, 145 cyclic group, 237 cyclic shift, 145 Data Encryption Standard, 202 decision rule, 91 decryption functions, 172 DES, 202 designed distance, 156 dictionary, 67 dictionary order, 62 Diffie-Hellman system, 230 digram, 164 dimension, 124 Discrete Logarithm Problem, 226 DLP, 226 double-locking, 204 ElGamal cryptosystem, 228 ElGamal signature scheme, 234 elliptic curve, 241 encoded stream, 13, 90 encryption, 171 encryption functions, 171 english, 163 entropy, 32

272

entropy of stationary source, 55 error-correcting code, 101 Euclidean algorithm, 212 exhaustive search, 173 extended BSC, 94 extended channel, 94 Fano’s inequality, 117 Feistel iteration, 199 final stream, 91 frequency analysis, 175 frequency table, 164 generator (of a cyclic group), 237 Gilbert-Varshamov bound, 105 Hamming code, 141 Hamming distance, 95 hash function, 222 Hasse’s theorem, 247 Hill’s system, 186 Huffman’s rule, 40 ideal, 147 ideal observer rule, 96 independent, 49 index, 67 index of coincidence, 182 information, 35 information rate, 104 integrity, 221 inverse probabilities, 87 key, 171 key distribution problem, 204 key equivocation, 193 keyword, 175 known ciphertext attack, 187 known plaintext attack, 187 Kraft-McMillan number, 18 left-inverse, 172 length, 4 linear code, 124 LZW encoding, 68 MAC, 223 marginal distributions, 49 maximum likelihood rule, 97 MD rule, 97 memoryless source, 29 message, 4 message authentication code, 223 message bits, 130

minimum distance, 100 minimum distance rule, 97 mistake, 91 mono-alphabetic substitution, 174 Morse code, 7 NAF, 250 neighbourhood, 100 noisy channel, 74 non-adjacent form, 250 non-repudiation, 221 one-time pad, 197 optimal code, 31 original stream, 13, 90 packing bound, 102 parity check, 128 password, 227 perfect code, 143 perfect secrecy, 195 PF, 15 phi function, 208 plaintext, 172 Playfair system, 183 point at infinity, 241 poly-alphabetic encryption, 180 prefix-free, 15 primality testing, 211 primitive, 155 primitive root, 225 private key, 207 probability distribution, 28 probability of a mistake, 108 product channel, 94 public key, 207 public key cryptography, 207 rate, 104 received stream, 90 redundancy, 169 repeated-squaring, 213 root, 16 RSA cryptosystem, 207 S-box, 202 semaphore, 8 SF rule, 39 Shannon’s theorem, 119 Shannon-Fano rule, 39 side-channel attacks, 220 signature, 232 signature functions, 232 signature scheme, 232

Index

source, 27 spurious key, 194 standard form, 130 stationary source, 53 stream, 13 string, 4 subspace, 123 symmetric key cryptosystem, 179 syndrome, 135 syndrome look-up table, 137 ternary, 5 ternary asymmetric channel, 76 triangle inequality, 96

273

UD, 6 uncertainty, 34 uncertainty of a natural language, 166 unicity point, 194 Unicode, 3 uniquely decodable, 6 valid signature, 232 verification algorithm, 232 Vigen`ere system, 180 weight, 124 word, 4